Question intro-1
What are the numbers called that are multiples of both 3 and 5?
Steps only
List multiples: 3, 6, 9, 12, 15… and 5, 10, 15…
15 is the first number in both lists.
Answer
A common multiple is divisible by each of the given numbers.
CLASS 6 · GANITA PRAKASH · CHAPTER 5
Prime Time
Clear worked answers on common factors and multiples, prime numbers, co-primes, prime factorisation and divisibility.
108 worked answersAll 9 end-of-chapter questions
What you will learn
- Find and use common factors and multiples.
- Recognise primes and composites and factor numbers into primes.
- Use divisibility rules to solve number puzzles.
108 worked answers · short steps first, explanations when you need them
5.1 Common Multiples and Common Factors
Textbook pages 108–111
Question 1
At what number is “idli-vada” said for the 10th time?
Steps only
Both words are said at common multiples of 3 and 5.
Their common multiples are 15, 30, 45, …
The 10th is 10 × 15.
Answer
The common multiples of 3 and 5 are multiples of 15. Counting the first as 15, the tenth is ten groups of 15.
Question 2(a)
From 1 to 90, how many times is “idli” said, counting “idli-vada” too?
Steps only
Count multiples of 3: 90 ÷ 3 = 30.
Answer
Every multiple of 3 gets “idli”, including numbers that are also multiples of 5.
Question 2(b)
From 1 to 90, how many times is “vada” said, counting “idli-vada” too?
Steps only
Count multiples of 5: 90 ÷ 5 = 18.
Answer
Every multiple of 5 gets “vada”, including common multiples.
Question 2(c)
From 1 to 90, how many times is “idli-vada” said?
Steps only
Common multiples are multiples of 15.
90 ÷ 15 = 6.
Answer
A number is called “idli-vada” when divisible by both 3 and 5, hence by 15.
Question 3
How many times are each word said when the game goes to 900?
Steps only
Multiples of 3: 900 ÷ 3 = 300.
Multiples of 5: 900 ÷ 5 = 180.
Common multiples of 15: 900 ÷ 15 = 60.
Answer
Count the multiples of each number in 1–900. Common multiples are included in both individual counts as the question asks.
Question 4(a)
What do the common entries in the multiples-of-3 and multiples-of-5 diagram represent?
Steps only
A number in the overlap is divisible by 3 and by 5.
Answer
The overlap of two sets contains the values belonging to both sets.
Question 4(b)
Name the common multiples of 3 and 4 shown up to 60.
Steps only
List multiples of 3 and of 4 up to 60.
The overlap begins 12, 24, 36, 48, 60.
Answer
Numbers in both lists are common multiples; they are multiples of 12.
Question 4(c)
One game number was 4 and nobody ever said only “vada”. Which listed number could be the other game number?
Steps only
If the other number is 8, every multiple of 8 is also a multiple of 4.
So “vada” alone never occurs.
Answer
For a number to be only “vada”, it must be a multiple of the second number but not of 4. If the second number is a multiple of 4, this cannot happen.
Question 4(d)
Is the figure related to the idli-vada game, and what numbers trigger both words when the game goes up to 60?
Steps only
The game uses 3 for “idli” and 5 for “vada”.
Common multiples are multiples of 15.
Up to 60 they are 15, 30, 45 and 60.
Answer
Yes. Both words are said at common multiples of 3 and 5. Up to 60, mark 15, 30, 45 and 60.
Question jump-15-30
What jump sizes land exactly on both 15 and 30?
Steps only
Find the common factors of 15 and 30.
Factors of 15 are 1, 3, 5, 15; all four divide 30 too.
Answer
1, 3, 5 and 15.
Question jump-24
Which jump sizes land exactly on 24?
Steps only
A jump size must divide 24 with no remainder.
The factor pairs are 1×24, 2×12, 3×8 and 4×6.
Answer
1, 2, 3, 4, 6, 8, 12 and 24.
Question jump-14-36
What jump sizes land on both treasure numbers 14 and 36?
Steps only
Factors of 14 are 1, 2, 7, 14.
Factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18, 36.
Keep the factors that appear in both lists.
Answer
Jump sizes 1 and 2 land on both 14 and 36.
Question multiples-chart-3
What is common to the shaded numbers in the chart from 31 to 70?
Steps only
Check which numbers divide each shaded entry.
The shaded entries are multiples of 3.
Answer
Every shaded number is a multiple of 3.
Question multiples-chart-4
What is common to the circled numbers in the chart from 31 to 70?
Steps only
Check which numbers divide each circled entry.
The circled entries are multiples of 4.
Answer
Every circled number is a multiple of 4.
Question multiples-chart-common
Which numbers from 31 to 70 are both shaded and circled, and what are they called?
Steps only
List values divisible by both 3 and 4.
These are 36, 48 and 60.
Answer
36, 48 and 60. They are common multiples of 3 and 4.
Question 5
Find multiples of 40 between 310 and 410.
Steps only
Count by 40: 280, 320, 360, 400, 440.
Keep only those strictly between 310 and 410.
Answer
A multiple is found by multiplying 40 by a whole number; then check the requested interval.
Question 6(a)
I am less than 40, have 7 as a factor, and my digits add to 8. Who am I?
Steps only
Multiples of 7 below 40: 7, 14, 21, 28, 35.
Only 35 has digits summing to 8.
Answer
Check both clues: 35 is a multiple of 7 and 3+5=8.
Question 6(b)
I am less than 100, divisible by both 3 and 5, and my digits differ by 1. Who am I?
Steps only
A number divisible by 3 and 5 is a multiple of 15.
45 is below 100 and its digits differ by 1.
Answer
Multiples of 15 below 100 include 15, 30, 45, 60, 75 and 90; only 45 has digits differing by 1.
Question 7
Find a perfect number between 1 and 10.
Steps only
The factors of 6 are 1, 2, 3 and 6.
Add them: 1+2+3+6=12=2×6.
Answer
A perfect number has the sum of all its positive factors equal to twice the number.
Question 8(a)
Find the common factors of 20 and 28.
Steps only
Factors of 20: 1, 2, 4, 5, 10, 20.
Factors of 28: 1, 2, 4, 7, 14, 28.
Answer
A common factor divides both numbers exactly.
Question 8(b)
Find the common factors of 35 and 50.
Steps only
Factors of 35: 1, 5, 7, 35.
Factors of 50: 1, 2, 5, 10, 25, 50.
Answer
Compare the factor lists and keep the values present in both.
Question 8(c)
Find the common factors of 4, 8 and 12.
Steps only
List factors: 4 → 1,2,4; 8 → 1,2,4,8; 12 → 1,2,3,4,6,12.
Answer
A common factor must divide every number in the group.
Question 8(d)
Find the common factors of 5, 15 and 25.
Steps only
The factors of 5 are 1 and 5.
Both 1 and 5 divide 15 and 25 too.
Answer
Every common factor must in particular be a factor of 5, so there can be no others.
Question 9
Give three multiples of 25 that are not multiples of 50.
Steps only
Multiples of 25 alternate between odd and even multiples: 25, 50, 75, 100, 125, …
Choose odd multiples of 25.
Answer
Odd multiples of 25 are not divisible by 50; many valid sets are possible.
Question 10
Two different numbers below 10 have their first common multiple after 50. Give possible pairs.
Steps only
For 7 and 8, the first common multiple is 56.
For 8 and 9, it is 72.
Answer
The first common multiple is the least common multiple. These pairs have least common multiples greater than 50.
Question 11
Which jump sizes land on both 28 and 70?
Steps only
A jump size must divide both target numbers.
Common factors of 28 and 70 are 1, 2, 7 and 14.
Answer
Landing exactly on a target means the jump size divides that target.
Question 12
The overlap shows common multiples of two numbers as 24, 48 and 72. What could the two numbers be?
Steps only
These values are multiples of 24.
One possible factor pair is 3 and 8; their common multiples include 24, 48 and 72.
Answer
The overlap alone can allow more than one pair. Check each candidate against both lists and the non-overlap entries in the full diagram.
Question 13
Find the smallest number that is a multiple of every number from 1 to 10 except 7.
Steps only
Take the greatest needed prime powers: 2³, 3² and 5.
Multiply: 8×9×5=360.
Answer
A number divisible by 1 through 10 except 7 needs the highest powers of 2, 3 and 5 appearing in those numbers.
Question 14
Find the smallest number that is a multiple of every number from 1 to 10.
Steps only
Use the greatest prime powers needed: 2³ (from 8), 3² (from 9), 5 and 7.
Multiply: 8×9×5×7.
Answer
To be divisible by every number, include enough copies of each prime to cover the largest power in 1–10.
5.2 Prime Numbers
Textbook pages 112–114
Question 1(a)
How many primes and composite numbers are there from 21 to 30?
Steps only
Primes are 23 and 29.
The other eight numbers in the interval each have a factor besides 1 and themselves.
Answer
A prime has exactly two positive factors; a composite number has more than two.
Question 1(b)
Is there an even prime other than 2?
Steps only
Every even number is divisible by 2.
If it is greater than 2, it has factors 1, 2 and itself.
Answer
A prime can have only two positive factors, so any even number greater than 2 is composite.
Question 2
What are the smallest and largest differences between successive primes up to 100?
Steps only
The smallest gap is 3−2=1.
The largest gap is 97−89=8.
Answer
Subtract each prime from the next one; the gap can vary.
Question 3
Do all rows of ten numbers in the 1–100 chart contain the same number of primes? Which rows have the fewest and most?
Steps only
Count primes in each decade row.
1–10 and 11–20 each have four; 91–100 has one.
Answer
The sieve or a prime list can be used to count primes decade by decade.
Question 4
Which of 23, 51, 37 and 26 are prime?
Steps only
23 has no divisor among primes up to √23; 37 has none up to √37.
51=3×17 and 26=2×13.
Answer
To test a number, check divisibility by primes no greater than its square root.
Question 5
Give three pairs of primes below 20 whose sum is a multiple of 5.
Steps only
Examples: 2+3=5; 3+7=10; 2+13=15.
Answer
Any pair of primes below 20 with a sum divisible by 5 works.
Question 6
Find reversed-digit prime pairs up to 100 like 13 and 31.
Steps only
Reverse digits of two-digit primes and test both numbers.
Answer
Both a number and its reversal must be prime.
Question 7
Find seven consecutive composite numbers between 1 and 100.
Steps only
Use 90 through 96.
They are divisible respectively by 2, 7, 2, 3, 2, 5 and 2 (with factors other than 1 and itself).
Answer
A run is verified by showing each number has a non-trivial factor.
Question 8
List the twin-prime pairs below 100.
Steps only
Check consecutive primes for a difference of 2.
Answer
Twin primes are prime numbers whose difference is exactly 2.
Question 9(a)
Is it true that no prime ends in 4?
Steps only
Any number ending in 4 is even and greater than 2.
It is divisible by 2, so it is composite.
Answer
2 is the only even prime; all numbers ending in 4 are even and greater than 2.
Question 9(b)
Can a product of primes be prime?
Steps only
The product of two primes is divisible by each factor.
Answer
It has factors besides 1 and the product, for example 2×3=6.
Question 9(c)
Do prime numbers have no factors?
Steps only
A prime has two factors: 1 and itself.
Answer
For 7, the factors are 1 and 7.
Question 9(d)
Are all even numbers composite?
Steps only
2 is even and has just factors 1 and 2.
Every larger even number has 2 as an extra factor.
Answer
8, for example, has factor 2 as well as 1 and 8.
Question 9(e)
For every prime except 2, is the next number composite?
Steps only
Every prime other than 2 is odd.
The next integer is even and greater than 2.
Answer
An even number greater than 2 has 2 and itself as distinct factors, plus 1.
Question 10
Which of 45, 60, 91, 105 and 330 is a product of exactly three different primes?
Steps only
Factor each: 105=3×5×7.
The others either repeat a prime or have a different number of prime factors.
Answer
“Distinct” means no prime is repeated; the number must have exactly three prime factors.
Question 11
Can the digits 2, 4 and 5 make a three-digit prime when each is used once?
Steps only
A number ending in 2 or 4 is even.
A number ending in 5 is divisible by 5.
Answer
A prime greater than 5 cannot end in an even digit or 5.
Question 12
Give five primes p for which 2p+1 is also prime.
Steps only
Try p=5, 11, 23, 29 and 41.
The resulting numbers are 11, 23, 47, 59 and 83.
Answer
This is an exploration: each proposed example must be checked for primality.
5.3 Co-prime Numbers
Textbook pages 115–116
Question 1(a)
Can one jump size land on both 15 and 39?
Steps only
Common jump sizes are common factors.
15 and 39 share 1 and 3.
Answer
A jump size can land on a target exactly when it divides that target.
Question 1(b)
Can one jump size land on both 4 and 15?
Steps only
The factors of 4 are 1,2,4; those of 15 are 1,3,5,15.
Only common factor is 1.
Answer
Here “safe” means there is no common jump larger than 1; jump 1 always reaches both.
Question 1(c)
Can one jump size land on both 18 and 29?
Steps only
29 is prime and does not divide 18.
Their only common factor is 1.
Answer
Two numbers are co-prime when their greatest common factor is 1.
Question 1(d)
Can one jump size land on both 20 and 55?
Steps only
Both are divisible by 5.
Answer
A common factor greater than 1 is a jump that reaches both treasures.
Question 2(a)
Are 18 and 35 co-prime?
Steps only
18=2×3² and 35=5×7.
There is no shared prime factor.
Answer
Numbers without a common factor other than 1 are co-prime.
Question 2(b)
Are 15 and 37 co-prime?
Steps only
15=3×5; 37 is prime and not 3 or 5.
Answer
Prime factorisation makes common factors easy to spot.
Question 2(c)
Are 30 and 415 co-prime?
Steps only
30=2×3×5; 415=5×83.
They share factor 5.
Answer
A shared prime factor means the numbers have a common factor greater than 1.
Question 2(d)
Are 17 and 69 co-prime?
Steps only
17 is prime; 69=3×23.
There is no common prime factor.
Answer
Check the factor lists for a shared factor other than 1.
Question 2(e)
Are 81 and 18 co-prime?
Steps only
81=3⁴ and 18=2×3².
They share factor 3.
Answer
A common prime factor proves a pair is not co-prime.
Question 3
When is the first common multiple equal to the product of two numbers? Give examples.
Steps only
For co-prime numbers, no prime factor is shared, so their least common multiple uses all factors of both.
Examples: 3 and 5 give 15; 3 and 7 give 21.
Answer
Prime factors shared by both numbers need only be included once in the least common multiple.
Question co-prime-art
In the thread art, does the thread visit every peg before returning to its start? How is that related to the number of pegs and the thread-gap?
Steps only
Repeatedly add the gap to the current peg, wrapping around after the final peg.
For 12 pegs and gap 4, the thread returns after 3 pegs; it makes separate small loops.
For 13 pegs and gap 3, it reaches every peg because 13 and 3 are co-prime.
Answer
When the two numbers share a factor greater than 1, the thread returns early and forms separate loops. For example, 12 pegs with gap 4 makes loops of 3 pegs; 13 pegs with gap 3 visits all 13 pegs. The other diagrams show the same co-prime pattern.
5.4 Prime Factorisation
Textbook pages 120–122
Question 1(a)
Find the prime factorisation of 64.
Steps only
Divide repeatedly by 2: 64→32→16→8→4→2→1.
Answer
A prime factorisation writes a number as a product of primes only.
Question 1(b)
Find the prime factorisation of 104.
Steps only
104=2×52=2×2×26=2×2×2×13.
Answer
Continue factoring each composite factor until only primes remain.
Question 1(c)
Find the prime factorisation of 105.
Steps only
105=3×35=3×5×7.
Answer
105=3×5×7.
Question 1(d)
Find the prime factorisation of 243.
Steps only
243=3×81=3×3×27=3×3×3×9=3×3×3×3×3.
Answer
243=3⁵.
Question 1(e)
Find the prime factorisation of 320.
Steps only
320=32×10=2⁵×2×5.
Answer
320=2⁶×5.
Question 1(f)
Find the prime factorisation of 141.
Steps only
1+4+1=6, so 141 is divisible by 3.
141÷3=47, which is prime.
Answer
141=3×47.
Question 1(g)
Find the prime factorisation of 1728.
Steps only
1728=2⁶×27.
27=3³.
Answer
1728=2⁶×3³.
Question 1(h)
Find the prime factorisation of 729.
Steps only
Repeatedly divide by 3: 729→243→81→27→9→3→1.
Answer
729=3⁶.
Question 1(i)
Find the prime factorisation of 1024.
Steps only
Divide by 2 repeatedly ten times.
Answer
2¹⁰=1024.
Question 1(j)
Find the prime factorisation of 1331.
Steps only
11×11=121 and 121×11=1331.
Answer
1331=11³.
Question 1(k)
Find the prime factorisation of 1000.
Steps only
1000=10×10×10 and 10=2×5.
Answer
1000=2³×5³.
Question 2
A number has prime factors one 2, two 3s and one 11. Find it.
Steps only
Multiply the factors: 2×3×3×11.
2×9×11=198.
Answer
The prime factors tell us exactly which factors to multiply.
Question 3
Find three primes below 30 whose product is 1955.
Steps only
1955 ends in 5, so divide by 5: 1955÷5=391.
391=17×23.
Answer
Factor the product step by step and check that each factor is prime.
Question 4(a)
Find the prime factorisation of 56×25 without multiplying first.
Steps only
56=2³×7 and 25=5².
Combine the prime factors.
Answer
Factor each part first; then join the prime factors.
Question 4(b)
Find the prime factorisation of 108×75 without multiplying first.
Steps only
108=2²×3³; 75=3×5².
Combine: 2²×3⁴×5².
Answer
2²×3⁴×5².
Question 4(c)
Find the prime factorisation of 1000×81 without multiplying first.
Steps only
1000=2³×5³; 81=3⁴.
Answer
2³×3⁴×5³.
Question 5(a)
What is the smallest number whose prime factorisation contains three different primes?
Steps only
Use the three smallest primes: 2, 3 and 5.
Multiply 2×3×5.
Answer
To make the smallest such product, choose the three smallest distinct primes.
Question 5(b)
What is the smallest number whose prime factorisation contains four different primes?
Steps only
Use 2, 3, 5 and 7.
Multiply 2×3×5×7.
Answer
Any four distinct primes must include at least these four smallest primes.
Question 6(a)
Are 30 and 45 co-prime?
Steps only
30=2×3×5; 45=3²×5.
They share 3 and 5.
Answer
A common prime factor means there is a common factor greater than 1.
Question 6(b)
Are 57 and 85 co-prime?
Steps only
57=3×19; 85=5×17.
No prime factor is shared.
Answer
Yes.
Question 6(c)
Are 121 and 1331 co-prime?
Steps only
121=11²; 1331=11³.
Answer
No; both are divisible by 11.
Question 6(d)
Are 343 and 216 co-prime?
Steps only
343=7³; 216=2³×3³.
No prime factor is shared.
Answer
Yes.
Question 7(a)
Is 225 divisible by 27?
Steps only
225=3²×5²; 27=3³.
225 does not contain three factors of 3.
Answer
No.
Question 7(b)
Is 96 divisible by 24?
Steps only
96=2⁵×3; 24=2³×3.
All factors of 24 occur in 96.
Answer
Yes; 96÷24=4.
Question 7(c)
Is 343 divisible by 17?
Steps only
343=7³ and has no factor 17.
Answer
No.
Question 7(d)
Is 999 divisible by 99?
Steps only
999=3³×37; 99=3²×11.
The factor 11 needed for 99 is missing.
Answer
No.
Question 8
Numbers have prime factorizations 2×3×7 and 3×7×11. Are they co-prime? Does either divide the other?
Steps only
Both share 3 and 7, so they are not co-prime.
The first has 2 but not 11; the second has 11 but not 2.
Answer
Divisibility requires every prime factor, with its full power, of the divisor to occur in the other number.
Question 9
Are any two different prime numbers co-prime?
Steps only
Different primes have no common factor besides 1.
Answer
A prime’s only factors are 1 and itself, and the two primes are different.
5.5 Revisiting the Test for Divisibility
Textbook pages 124–125
Question intro-1
Is 8536 divisible by 4?
Steps only
The last two digits are 36.
36÷4=9.
Answer
A number is divisible by 4 exactly when its last two digits form a multiple of 4.
Question factors-of-8560
Which of 2, 3, 4, 5, 6, 7, 8, 9 and 10 divide 8560 exactly?
Steps only
8560 is even, ends in 0, and its last two digits 60 divide by 4.
Its last three digits 560 divide by 8.
Its digit sum is 19, and it is not divisible by 7.
Answer
8560 is divisible by 2, 4, 5, 8 and 10. It is not divisible by 3, 6, 7 or 9.
Question intro-2
Why do only the last two digits decide divisibility by 4?
Steps only
Every hundred is divisible by 4, since 100=25×4.
All hundreds before the last two digits contribute a multiple of 4.
Answer
For example, 8536=8500+36; both 8500 and 36 are divisible by 4.
Question divisibility-by-4-exploration
Why can the final digit alone not decide divisibility by 4, and what multiples of 4 lie in three sample intervals?
Steps only
12 and 22 end in 2, but only 12 is divisible by 4.
Using the final two digits, the strict intervals give 332,336; 1732,1736; and 2032,2036.
Answer
The unit digit is not enough: 12 is divisible by 4 but 22 is not. In the open intervals, the multiples are 332, 336; 1732, 1736; and 2032, 2036. Include 1740 or 2040 if the endpoint is counted.
Question divisibility-by-8-rule
Can the last two digits decide divisibility by 8? Are these statements correct: only the last three digits matter; if those digits form a multiple of 8 then the whole number does; if the whole number is divisible by 8 then its last three digits are?
Steps only
100 is not divisible by 8, so the last two digits alone do not always work.
1000 is divisible by 8. Write a number as a multiple of 1000 plus its final three-digit part.
Answer
A thousand is divisible by 8, so the digits before the final three contribute a multiple of 8. Therefore the full number and its last three digits have the same remainder when divided by 8.
Question explore-10-5-2
Use the patterns in multiples to decide: does 125 divide by 10; is 8560 divisible by 10 and 5; what is the greatest multiple of 5 below 399; is 682 divisible by 2; and which numbers from 399 to 411 are divisible by 2?
Steps only
Numbers divisible by 10 end in 0, so 125 is not divisible by 10 and 8560 is.
Numbers divisible by 5 end in 0 or 5; the greatest below 399 is 395.
682 and 8560 are even, so both divide by 2.
From 399 to 411, list the even numbers.
Answer
125 is not divisible by 10; 8560 is divisible by both 10 and 5; the largest multiple of 5 below 399 is 395; 682 and 8560 are divisible by 2; the even numbers in the interval are 400, 402, 404, 406, 408 and 410.
Question multiples-of-4-exploration
Which numbers between 330 and 340, 1730 and 1740, and 2030 and 2040 are divisible by 4?
Steps only
Check the final two digits.
For the strict intervals, the values are 332, 336; 1732, 1736; and 2032, 2036.
Answer
Using “between” to exclude the endpoints, the answers are 332 and 336; 1732 and 1736; 2032 and 2036. The endpoint 1740 or 2040 is also divisible by 4 if the book’s interval is read inclusively.
Question 1
Find numbers divisible by 8 between 120 and 140, 1120 and 1140, and 3120 and 3140.
Steps only
Between 120 and 140: 128, 136.
The same last three digits give 1128,1136 and 3128,3136.
Answer
Adding a multiple of 1000 does not change divisibility by 8 because 1000 is divisible by 8.
Question 2
Change the last two digits of 8560 to make a multiple of 8. Give one answer.
Steps only
Keep 85 hundreds and choose a last-two-digit number so the whole number divides by 8.
8552÷8=1069.
Answer
Many endings work; test the complete number by dividing by 8.
Question 3(a)
How many leap years are there from 2024 through 2099?
Steps only
Years divisible by 4 in this interval: 2024, 2028, …, 2096.
There are (2096−2024)÷4+1=19 years.
None is a century year.
Answer
Count multiples of 4 inclusively; the century exception does not apply within this interval.
Question 3(b)
Which leap years have occurred from your birth year to now?
Steps only
List years divisible by 4 in your personal date range.
If a year ends in 00, keep it only if divisible by 400.
Answer
The answer depends on the student’s birth year; apply the century rule as well as divisibility by 4.
Question 4
Find the smallest and largest four-digit palindromes divisible by 4.
Steps only
A four-digit palindrome has form ABBA. Its last two digits are BA.
The first digit A must be even for BA to be even.
The smallest possible is 2112; the largest is 8888.
Answer
Test divisibility using the last two digits: 12 and 88 are divisible by 4. Palindromes 1001 and 9999 are not divisible by 4.
Question 5(a)
Is the sum of two even numbers always a multiple of 4?
Steps only
2+6=8 is divisible by 4.
2+4=6 is not divisible by 4.
Answer
Examples with different remainders modulo 4 show the result is not guaranteed.
Question 5(b)
Is the sum of two odd numbers always a multiple of 4?
Steps only
1+3=4 is divisible by 4.
1+5=6 is not divisible by 4.
Answer
Two odd numbers make an even sum, but that even sum may or may not be divisible by 4.
Question 6
Find remainders on division by 10, 5 and 2 for 78, 99, 173, 572, 980, 1111 and 2345.
Steps only
For 10, read the units digit.
For 5, use the remainder of the final digit when divided by 5.
For 2, even numbers leave 0; odd numbers leave 1.
Answer
The units digit determines remainders for 10, 5 and 2.
Question 7
Which two divisibility checks are enough to establish that 14560 is divisible by 2, 4, 5, 8 and 10?
Steps only
Check divisibility by 5: it ends in 0.
Check divisibility by 8: 560÷8=70.
Divisibility by 8 guarantees divisibility by 2 and 4; ending in 0 also gives divisibility by 10.
Answer
A number divisible by 8 is also divisible by 2 and 4. A number ending in zero is divisible by 5 and 10.
Question 8
Which of 572, 2352, 5600, 6000 and 77622160 are divisible by all of 2,4,5,8 and 10?
Steps only
A number divisible by 10 must end in 0.
Among those, test the last three digits for divisibility by 8.
Answer
Divisibility by 8 ensures divisibility by 2 and 4; ending in 0 ensures divisibility by 5 and 10.
Question 9
Give two factors of 10000 whose product is 10000 and neither ends in 0.
Steps only
16×625=10000.
Answer
The task asks for a factor pair avoiding a units digit of zero.
5.6 Fun with Numbers
Textbook pages 126–128
Question 1
Which of 9, 16, 25 and 43 could be called special, and why might students choose different numbers?
Steps only
9 is the only one-digit number and the only multiple of 3.
16 is the only even number and the only multiple of 4.
25 is the only multiple of 5.
43 is the only prime and the only number that is not a square.
Answer
All four can be called special for different reasons: 9 is one digit and divisible by 3; 16 is even and divisible by 4; 25 is divisible by 5; 43 is prime and not a square. A choice is valid when its property distinguishes it from the others.
Question 1(b)
Give possible reasons that the numbers in each four-number box are special.
Steps only
Box 1 (5, 7, 12, 35): 5 and 7 are prime; 12 is even; 35 is a multiple of 5 and 7.
Box 2 (3, 8, 11, 24): 3 and 11 are prime; 8 is a power of 2; 24 is divisible by 3 and 8.
Box 3 (27, 3, 123, 31): 27 is a cube; 3 and 31 are prime; 123 is a three-digit multiple of 3.
Box 4 (17, 27, 44, 65): 17 is prime; 27 is a cube; 44 is even; 65 is a multiple of 5.
Answer
These are examples of how to describe each value by a property. Students may find other correct properties; they should explain and check each one.
Question 2
Fill the two 3-by-3 prime grids so row products are 105, 20, 30 and 8, 105, 70; the column products are 28, 125, 18 and 30, 70, 28.
Steps only
Factor every row and column product into primes.
Grid 1 rows: 7,5,3 / 2,5,2 / 2,5,3.
Grid 2 rows: 2,2,2 / 3,5,7 / 5,7,2.
Multiply each row and column to check.
Answer
One valid completion is Grid 1: [7,5,3; 2,5,2; 2,5,3]. Grid 2: [2,2,2; 3,5,7; 5,7,2]. Check row and column products against the labels.
Question 3
What is true about the prime factorisation of a whole number greater than 1?
Steps only
Keep splitting composite factors into smaller factors until all are prime.
Answer
This is the fundamental prime-factorisation property explored in the chapter.