Factors and Multiples

November 23, 2022

This is a set of word problems that have to do with factors or multiple of two or more numbers.

There are two strings of 12cm and 18cm, respectively. You have to cut them into pieces so that there is no extra string left over, and each piece should be the largest length possible.

A shopkeeper sells candles in packets of 12 and candle stands in a packet of 8. What is the least number of candles and candle stands Nita should buy so that there will be one candle for each candle stand?

I swim every 6 days, run every 4 days, and cycle every 15 days. I did all activities today; when is the next time I will do them again on the same day?

All of these and more require the student to understand the concept of multiples and factors.

Prerequisite concepts

Prime numbers

A prime number is divisible by 1 and itself.
- The first few primes are 2, 3, 5, 7, 13, 17, 19, …

Rules of divisibility

For 2, 3, 4, 5, 6, 9

Multiples

Multiples of 3

3, 6, 9, 12, 15, 18, ...

Multiples of 4

4, 8, 12, 16, 20, 24, 28, ...

So a prerequisite for this is to know the tables well…or well enough, at least.
The multiples are always bigger than the number whose multiple is being listed.
Two numbers may have many common multiples.
The lowest common multiple (LCM) is special; for example, it solves the swimming problem above.

LCM

If the number is less than 20 then just write out the first few multiples
- Choose the smallest one, and that’s the LCM.
Using the prime factorisation method
Using the division method

Factors

Factors of 12

1 x 12 = 12
2 x 6 = 12
3 x 4 = 12

Factors: 1, 2, 3, 4, 6, 12

Factors of 18

1 x 18 = 18
2 x 9 = 18
3 x 6 = 18

Factors: 1, 2, 3, 6, 9, 18

You start with 1 and go to 9, trying to find pairs of numbers whose product will be the number whose factor you want. This works well for small numbers of course.

Factors are always smaller than the numbers whose factors you seek.
Two numbers may have many common factors.
Often, the highest common factor (HCF) is special; it gives you the largest number you can divide both numbers by.

Prime factorisation

https://www.youtube.com/watch?v=DNAqY8YucGI

Successive Division

https://www.youtube.com/watch?v=eLX5QRVBHQ0

HCF

Use the above method to list all the factors, then pick out the largest common one.
- For 12 and 18, the HCF is: 6
Use the prime factorization method
Use the division method