Maharashtra Board Class 5 Maths Part Two Chapter 8 Multiples and Factors PDF Download

8. Multiples And Factors

Dada: I have 12 pedhas. I have to put them in equal groups so that no pedha is left over. How many pedhas must I put in each group?

Sanju: Putting into groups means dividing. None left over means that the remainder will be 0.

Anju: 12 is divisible by 2, so we can make groups of two.

Manju: 12 is divisible by 3, so we can make groups of 3.

Sagar: We can also make groups of four.

Anita: Can we make groups of five?

Manju: No, because 12 is not divisible by 5.

Anju: We can divide 12 by six, so groups of six can be made.

Manju: We cannot make groups of 7, 8, 9, 10 or 11 because we cannot divide 12 by any of these numbers.

Sanju: We could make one group of twelve and give it to one person. Or, we could give 12 people 1 pedha each.

Dada: Very good. 12 is exactly divisible by 1, 2, 3, 4, 6 and 12, which means the remainder is 0. These numbers are called divisors or factors of 12. Similarly, 1, 2, 4, 8 and 16 are factors of 16.

Problem Set 32

Write the factors of the following numbers.

(1) 8 (2) 5 (3) 14 (4) 10 (5) 7 (6) 22 (7) 25 (8) 32 (9) 33

Teacher's Note

Factors are numbers that divide another number completely. Just like how your mother divides 12 sweets equally among 3 children without any left over - 3 is a factor of 12.

Exam Trick

Remember: Factor means "divides completely with no remainder." If you can divide a number perfectly, that number is a factor. Like 2 is a factor of 8 because 8 ÷ 2 = 4 with no leftover.

Points To Remember

Factors are numbers that divide another number exactly with remainder 0.
Every number has at least 2 factors: 1 and the number itself.
You can find factors by dividing the number by different numbers.
If a number divides perfectly, it is a factor.

Multiples

Dada: You know what a divisor and a dividend is. Do you know what a multiple is?

Anju: I don't know what a multiple is, but I think it must be related to multiplication.

Dada: Right! Let me give you an example. You can solve 20 ÷ 5, can't you?

Anju: Yes. When we divide the dividend 20 by the divisor 5, the quotient is 4 and the remainder is 0.

Dada: When the division of a dividend leaves no remainder, the dividend is said to be a multiple of the divisor. In such a case, the dividend is the product of the divisor and the quotient. Here, 20 is a multiple of 5, but 21 is not. Now tell me, can we divide 84 chalksticks into groups of six?

Suraj: Let me divide by 6. 84 can be divided exactly by 6 and the quotient is 14. Thus, we can make 14 groups of 6. So, 84 is the multiple of 6 and 6 is a factor of 84.

Teacher's Note

A multiple is a number that comes in a counting pattern. For example, 5, 10, 15, 20 are multiples of 5 - just like the 5 times table in school.

Exam Trick

Remember: Multiple = result of multiplication. If you multiply 5 × 3 = 15, then 15 is a multiple of 5. Just count in the 5 times table: 5, 10, 15, 20, 25...

Points To Remember

A multiple is a number that is exactly divisible by another number.
Multiples are found by multiplying a number by 1, 2, 3, 4 and so on.
Every number is a multiple of 1 and itself.
A number can have many multiples.

Dada: If the number of chalksticks is 6, 12, 18, 36 or 84, then we can make exact groups of 6 with none left over. It means that 6, 12, 18, 36 and 84 are multiples of 6, or that they are exactly divisible by 6. To see whether the number of chalksticks is a multiple of 6, divide that number by 6. If the remainder is 0, the number is a multiple of 6.

Each number in the 3 times table is exactly divisible by 3 or is a multiple of 3. Similarly, the numbers in the 7 times table are multiples of 7. Numbers in the 9 times table are multiples of 9.

We use this idea all the time. Let me ask you a few questions so as to make it clear. I have a 200 ml measure. Will I be able to measure out 1 litre of milk with it?

Suraj: I litre is 1000 ml. 1000 = 200 × 5, which means that 1000 is a multiple of 200. So we can measure out 1 litre of milk with the 200 millilitre measure. 5 measures of 200 ml make 1 litre.

Dada: Can we measure out one and a half litres of milk with the 200 ml measure?

Anju: One and a half litres is 1500 ml. 1500 is not divisible by 200. So, it is not a multiple of 200. So the 200 ml measure cannot be used to measure out one and a half litres of milk.

Dada: I have 400 grams of chana. I have to make pouches of 60 grams each. Is that possible, if I don't want any left overs?

Anju: No. 400 is not a multiple of 60.

Dada: How much more chana will I need to make those pouches of 60 grams each?

Anju: We will have to find the multiple of 60 that comes directly after 400. 60 × 6 = 360, 60 × 7 = 420. So, we need 20 grams more of chana.

Tests For Divisibility

Study the 2 times table and see which numbers appear in the units place. Similarly, divide 52, 74, 80, 96 and 98 by 2 to see if they are exactly divisible by 2. What rule do we get for determining whether a number is a multiple of 2?

Now study the 5 and 10 times tables.

See what rules you get for finding multiples of 5 and 10, that is, numbers divisible by 5 and 10.

Test for divisibility by 2: If there is 0, 2, 4, 6 or 8 in the units place, the number is a multiple of 2, or is exactly divisible by 2.

Test for divisibility by 5: Any number with 5 or 0 in the units place is a multiple of 5 or, is divisible by 5.

Test for divisibility by 10: Any number that has 0 in the units place is a multiple of 10.

Teacher's Note

Divisibility tests help us quickly check if a number is divisible by another. Like checking if 45 is divisible by 5 - it ends in 5, so yes! No need to divide.

Exam Trick

Remember: Look at the last digit (units place) to test divisibility by 2, 5, or 10. If a number ends in 0, 2, 4, 6, or 8, it's divisible by 2. If it ends in 5 or 0, it's divisible by 5.

Points To Remember

Even numbers (ending in 0, 2, 4, 6, 8) are divisible by 2.
Numbers ending in 5 or 0 are divisible by 5.
Numbers ending in 0 are divisible by 10.
You only need to look at the last digit for these tests.

Problem Set 33

1. (1) Write five three-digit numbers that are multiples of 2.

(2) Write five three-digit numbers that are multiples of 5.

(3) Write five three-digit numbers that are multiples of 10.

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

3. A 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.

4. A 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?

5. If the number given in the table is divisible by the given divisor, put a tick in the box. If it is not divisible by the divisor, put a cross in the box.

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