What is factor/divisor of a number? Why to find factor pairs only till square root of the number?

Divisor/Factor

A divisor or factor of a number perfectly divides that number (i.e without any remainder).

Example – 10/5 = 2 R0 -> 5 is a divisor/factor of 10

100/20 = 5 R0 -> 20 is a divisor/factor of 100

9/5 = 1 R4 -> 5 is not a divisor/factor of 9

You can perfectly divide 10 by 5 and 100 by 20. So 5 is a factor of 10 and 20 is a factor of 100. But 9/5 is not a whole number. You will end up with a decimal value on dividing 9 by 5. Hence, 5 is not a factor/divisor of 9.

How to find divisor/factor of a number(n)?

Starting from 1, you might have to check whether each and every number less than n is divisible by number. If you take 24, it is divisible by 1,2.. When you are checking that, it is obvious that along with 1 you need 24 to make it as 24(1*24). Similarly 2 has to be multiplied by 12 to make it as 24. Therefore it is time saving to write the number n in all possible pair of multiples..

To find out the factors of 24,

24 = 1*24 = 2*12 = 3*8 = 4*6 (It is not divisible by 5 and we already wrote  6 in 4*6. So we can stop)

The factors of 24 are 1,2,3,4,6,8,12 and 24.

Factors can also be negative. Hence -1,-2,-3,-4,-6,-8,-12 and -24 are also the factors of 24. We hardly apply negative factors in real time. So, we shall only consider positive factors going forward.

Lets try writing the factors of 100,

100 = 1*100

= 2*50

= 3*?? -> not possible, 3 is not a factor, lets move to 4

= 4*25

= 5*20

=6*?? -> 6*16 is 96. So 6 cannot be a divisor of 100

=7*?? -> not a divisor

= 8*?? -> again not possible

= 9*?? -> nope

=10*10 ( well we should only write 10 once in the list of factors though)

= 11*?? (no) = 12 *?? (no) = 13*??(no)….. = 20*5(hey we wrote that already. lets stop)

So the factors of 100 are 1,2,4,5,10,20,25,50 and 100

Why to stop at square root of the number

If you see the pairs of numbers that multiply to 100 – 1*100, 2*50, 4*25, 5*20, 10*10, you can observe that one of the numbers in the pair increases from 1 and the other one is decreasing from 100. This is happening until both the numbers are equal. If you try writing beyond 10 (square root of 100), you will find that the new factors that you come are already written down paired with smaller numbers. In other words, when a number is written as a product of two of its factors, one number will be less than or equal to its square root and the other will always be greater than or equal to its square root.

More explanation – You cannot write 100 as a multiple of two numbers greater than 10. If you try pairing it with a number greater than 10, the other number will be definitely smaller than 10. We have checked each and every number less than 10 and already paired it with the numbers greater than 10, so why bother checking numbers more than 10 if the numbers are covered already?

38 = 1 * 38 = 2 * 19 = 3 *?? (no) = 4* ??(no) = 5 *?? (no) = 6 *??(no) = 7 *??(stop here, Sq rt(36)=6.something. So don’t bother to check beyond 6. We have covered those already while pairing up with smaller numbers.) Hence, the factors of 38 are 1,2,19 and 38.

In general, if you write a number n as a product of two numbers a and b,

n=a*b

if a<=√n then b >=√n

If you are not very clear, write down the numbers -36, 66,81 (and as many more until you are convinced) as product of its all possible factor pairs. You will be clear about the pattern and can easily understand why this happens.

Some additional thoughts – You can also observe that, 1 and the number itself are always the factors of a number. Moreover, any perfect square number always have odd number of factors(In our example, 10*10=100, so we consider only one 10. Rest all factors are paired. So its always odd for perfect squares!) and non perfect square numbers always have even number of factors.