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How many integers less than 100 have an odd number of factors?
There are 9 integers less than 100 that have an odd number of factors.Every factor of a number has a pair, eg 2 & 3 are a factor pair of 6; and so it would be expected that every number has an even number of factors.However, if the factor pair of a number are the same number (eg 6 & 6 are a factor pair of 36), then there will be an odd number of factors.When there is a repeated factor like this, the number is a perfect square.Thus only perfect squares have an odd number of factors.Less than 100 there are 9 perfect squares (1, 4, 9, 16, 25, 36, 49, 64 & 81) which have an odd number of factors.100 itself is a perfect square and also has an odd number of factors, but the question asked for those numbers less than 100 with an odd number of factors.
Factors of 6?
Factors of the number 6 are 1,2,3,6
Does factor pairs mean one or two numbers?
A factor pair is two numbers. A factor pair is the set of two factors that when multiplied together result in the number of which they are factors. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factor pairs are 1 x 12, 2 x 6, and 3 x 4.
Why 81 have an odd number of factors and 40 have an even number of factors?
A number will have an odd number of factors if and only if one of those factors is the square root of the number. This is because any factor of a number you can find must pair up with another factor of that number, and multiply to give the number. The only exception is when the factor would be paired with itself. In other words, be the square root. 81 is a square number, and so the factor 9 (square root of 81) will not pair with any other factor, meaning 81 will have an odd number of factors. 40 has an even number of factors, because it is not a square number, and so all its factors can pair up.
Why is the set of factors of a number not the same as the set of proper factors of that number?
Proper factors of a number do not include 1 or the number itself. For example, all of the factors of 6 are 1, 2, 3, and 6, but the proper factors of 6 are 2 and 3.
