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Std 5 Mathematics · Chapter 8

8. Multiples and Factors

Solutions to Problem Sets 32 to 35 — factors, multiples, divisibility tests, and prime, composite and co-prime numbers. Maharashtra State Board (Balbharati), Grade Five Mathematics, Part Two.

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Key Ideas

Factor (divisor): a number that divides another number exactly, leaving remainder 0. The factors of 12 are 1, 2, 3, 4, 6, 12.
Multiple: if a number divides exactly by another, it is a multiple of that number. 84 ÷ 6 = 14, so 84 is a multiple of 6 and 6 is a factor of 84.
Divisibility tests: by 2 — units digit 0, 2, 4, 6 or 8; by 5 — units digit 0 or 5; by 10 — units digit 0.
Prime: exactly two factors, 1 and itself. Composite: more than two factors. 1 is neither prime nor composite.
Co-prime numbers: two numbers whose only common factor is 1.

Problem Set 32

QWrite the factors of the following numbers.
  1. Factors of 8: 1, 2, 4, 8
  2. Factors of 5: 1, 5
  3. Factors of 14: 1, 2, 7, 14
  4. Factors of 10: 1, 2, 5, 10
  5. Factors of 7: 1, 7
  6. Factors of 22: 1, 2, 11, 22
  7. Factors of 25: 1, 5, 25
  8. Factors of 32: 1, 2, 4, 8, 16, 32
  9. Factors of 33: 1, 3, 11, 33

Tip: factors come in pairs. For 32: 1 × 32, 2 × 16, 4 × 8. So the factors are 1, 2, 4, 8, 16, 32.

Problem Set 33

1(1) Write five three-digit numbers that are multiples of 2. (2) … multiples of 5. (3) … multiples of 10.
  1. Multiples of 2 (units digit 0, 2, 4, 6 or 8): 104, 236, 458, 670, 992
  2. Multiples of 5 (units digit 0 or 5): 105, 240, 375, 520, 995
  3. Multiples of 10 (units digit 0): 110, 250, 340, 780, 990

Many other answers are possible; check that the units digit follows the test for divisibility.

2Write 5 numbers that are multiples of 2 as well as of 3.

Numbers that are multiples of both 2 and 3 are the multiples of 6:

6, 12, 18, 24, 30

Check: 18 ÷ 2 = 9 and 18 ÷ 3 = 6, with no remainder in either case.

3A ribbon is 3 metres long. Can we cut it into 50 cm pieces and have nothing left over? Write the reason why or why not.

3 metres = 300 cm. 300 ÷ 50 = 6, remainder 0.

Yes, because 300 is a multiple of 50. The ribbon can be cut into 6 pieces of 50 cm each with nothing left over.

4A ribbon is 3 metres long. I need 8 pieces of ribbon each 40 cm long. How many centimetres shorter is the ribbon than the length I need?
Length needed: 8 × 40 = 320 cm
Length available: 3 m = 300 cm
Shortfall: 320 − 300 = 20 cm

The ribbon is 20 cm shorter than the length needed.

5If the number given in the table is divisible by the given divisor, put ✓ in the box. If it is not divisible by the divisor, put ✗ in the box.
Number \ Divisor2510
15✗✓✗
30✓✓✓
34✓✗✗
46✓✗✗
55✗✓✗
63✗✗✗
70✓✓✓
84✓✗✗

The row for 15 is the textbook example. Look only at the units digit: 30 and 70 end in 0, so they pass all three tests. 34, 46 and 84 are even but do not end in 0 or 5. 55 ends in 5, and 63 fails all three tests.

Prime and Composite Numbers

ASome numbers are given in the tables below. Write all of their factors.
NumberFactorsType
21, 2Prime
31, 3Prime
41, 2, 4Composite
51, 5Prime
61, 2, 3, 6Composite
111, 11Prime
121, 2, 3, 4, 6, 12Composite
161, 2, 4, 8, 16Composite
191, 19Prime
251, 5, 25Composite

The two textbook tables are combined here. The “Type” column is added to show which numbers are prime (only two factors) and which are composite (more than two factors).

Problem Set 34

1Write all the prime numbers between 1 and 20.

2, 3, 5, 7, 11, 13, 17, 19

There are 8 prime numbers between 1 and 20.

2Write all the composite numbers between 21 and 50.

22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49

The numbers from 22 to 49 are listed. The prime numbers 23, 29, 31, 37, 41, 43 and 47 are left out. If 21 and 50 are included, they are composite too (21 = 3 × 7, 50 = 5 × 10).

3Circle the prime numbers in the list given below: 22, 37, 43, 48, 53, 60, 91, 57, 59, 77, 79, 97, 100

Prime numbers (to be circled): 37, 43, 53, 59, 79, 97

Not prime: 22 = 2 × 11; 48 = 6 × 8; 60 = 6 × 10; 91 = 7 × 13; 57 = 3 × 19; 77 = 7 × 11; 100 = 10 × 10

Be careful with 91 and 57. They look prime, but 91 = 7 × 13 and 57 = 3 × 19.

4Which of the prime numbers are even numbers?

2 is the only even prime number. Every other even number is divisible by 2, so it has more than two factors.

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Problem Set 35

QDetermine whether the pairs of numbers given below are co-prime numbers.
  1. 22, 24 — common factors: 1, 2 → Not co-prime
  2. 14, 21 — common factors: 1, 7 → Not co-prime
  3. 10, 33 — common factors: 1 → Co-prime
  4. 11, 30 — common factors: 1 → Co-prime
  5. 5, 7 — common factors: 1 → Co-prime
  6. 15, 16 — common factors: 1 → Co-prime
  7. 50, 52 — common factors: 1, 2 → Not co-prime
  8. 17, 18 — common factors: 1 → Co-prime

Two consecutive numbers (such as 15, 16 or 17, 18) are always co-prime.

Activities

1Activity 1: Circle multiples of 2 in blue and multiples of 4 in red (1 to 60).
  • Do all numbers with a blue circle also have a red circle? No. For example, 2, 6, 10 and 14 have a blue circle but no red circle.
  • Do all the numbers with a red circle have a blue circle? Yes. Every number with a red circle (4, 8, 12, … 60) also has a blue circle.
  • Are all multiples of 2 also multiples of 4? No.
  • Are all multiples of 4 also multiples of 2? Yes, because 4 = 2 × 2.
2Activity 2: Draw a triangle around multiples of 2 and a circle around multiples of 3 (1 to 60). Find the numbers divisible by 6. What property do they share?

Numbers divisible by 6 from 1 to 60: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.

Property: every one of them has both a triangle and a circle. A number divisible by 6 is divisible by both 2 and 3, and a number divisible by both 2 and 3 is divisible by 6.

BEratosthenes’ Sieve — the prime numbers from 1 to 100.
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100

After striking out the multiples of 2, 3, 5 and 7, the numbers left (circled) are the prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

There are 25 prime numbers between 1 and 100. The number 1 is neither prime nor composite.

COne more method: write numbers up to 102 in six columns and find the prime numbers.

Column 1: 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97

Column 5: 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101

The prime numbers (in bold) other than 2 and 3 all lie in these two columns. The numbers in the columns for 2, 3, 4 and 6 are multiples of 2 or 3, so apart from 2 and 3 themselves none of them is prime. The two columns also contain some composite numbers, such as 25, 35, 49 and 91, which must still be checked.

DSomething more: find the eight pairs of twin prime numbers between 1 and 100.

3 and 5; 5 and 7; 11 and 13; 17 and 19; 29 and 31; 41 and 43; 59 and 61; 71 and 73

Each pair consists of two prime numbers with a difference of 2.

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