WHAT ARE THE DIVISORS OF 90? (LIST) - MATHS - 2025

The divisors of 90 are all those integers such that when dividing 90 by them the result is also a whole number.

In other words, an integer "a" is a divisor of 90 if when the division of 90 is made by "a" (90 ÷ a), the remainder of said division is equal to 0.

To find what the divisors of 90 are, we begin by decomposing 90 into prime factors.

Then, all possible products between those prime factors are realized. All results will be the divisors of 90.

The first divisors that can be added to the list are 1 and 90.

List of Divisors of 90

If all the divisors of the number 90 calculated above are grouped together, the set {1, 2, 3, 5, 6, 9, 15, 18, 30, 45} is obtained.

But, it must be remembered that the definition of the divisor of a number applies to whole numbers, that is, positive and negative. Therefore, to the previous set it is necessary to add the negative integers that also divide 90.

The calculations performed above could be repeated, but you can see that the same numbers will be obtained as before except that they will all be negative.

Therefore, the list of all divisors of the number 90 are:

{± 1, ± 2, ± 3, ± 5, ± 6, ± 9, ± 15, ± 18, ± 30, ± 45}.

Prime factors of 90

One detail to be careful with is that, when talking about divisors of a whole number, it is implicitly understood that the divisors must also be whole numbers.

That is, if you consider the number 3, you can see that dividing 3 by 1.5, the result will be 2 (and the remainder is equal to 0). But 1.5 is not considered a divisor of 3 as this definition is only for whole numbers.

By factoring 90 into prime factors, you can see that 90 = 2 * 3² * 5. Therefore, it can be concluded that both 2, 3 and 5 are also divisors of 90.

It remains to add all the possible products between these numbers (2, 3, 5), keeping in mind that 3 has a power of two.

Possible Products

So far, the list of divisors of the number 90 is: {1,2,3,5,90}. The other products to add are the products of only two integers, three integers, and four.

1.- Of two integers:

If the number 2 is set then the product takes the form 2 * _, the second place has only 2 possible options that are 3 or 5, therefore there are 2 possible products that involve the number 2, namely: 2 * 3 = 6 and 2 * 5 = 10.

If the number 3 is set then the product is of the form 3 * _, where the second place has 3 options (2, 3 or 5), but 2 cannot be chosen, since it was already chosen in the previous case. Therefore, there are only 2 possible products which are: 3 * 3 = 9 and 3 * 5 = 15.

If 5 is now set then the product takes the form 5 * _, and the options for the second integer are 2 or 3, but these cases have already been considered previously.

Therefore, there are a total of 4 products of two integers, that is, there are 4 new divisors of the number 90 which are: 6, 9, 10 and 15.

2.- Of three integers:

We start by setting 2 in the first factor, then the product is of the form 2 * _ * _. The different products of 3 factors with the number 2 fixed are 2 * 3 * 3 = 18, 2 * 3 * 5 = 30.

It should be noted that the product 2 * 5 * 3 has already been added. Therefore, there are only two possible products.

If 3 is set as the first factor, then the possible products of 3 factors are 3 * 2 * 3 = 18 (already added) and 3 * 3 * 5 = 45. Therefore, there is only one new option.

In conclusion, there are three new divisors of 90 which are: 18, 30 and 45.

3.- Of four integers:

If the product of four integers is considered then the only option is 2 * 3 * 3 * 5 = 90, which was already added to the list from the beginning.

References

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