How do you know that 144 has an even or odd number of factors without finding each one?

Answer 1

Express the number in its prime factorisation in power format.

The number of factors of the number is the product of the powers each increased by 1.

144 = 2^4 × 2^2 → number of factors is (4+1) × (2+1) = 3×5 = 15; so 144 has an odd number of factors.

As an even number times an even or odd number is even, the only way the product above can be odd is if every power increased by 1 is odd, that is only if every prime is to an even power.

By using the laws of indices, this means the power of every prime can be divided by 2 and the whole prime factorisation so created raised to the power 2 which is squared, eg for 144: 2^4 × 3^2 = (2^2 ×3^1)².

Thus for a number to have an odd number of factors it must be a perfect square.

144 is a perfect square (144 = 12²) therefore it has an odd number of factors.

This result that only perfect squares have an odd number of factors can also be seen by looking at the factor pairs of a number: the factors of a number can be paired up so that when the numbers of a pair are multiplied together they get the original number. If there is an odd number of factors then after the pairing there will be one factor left unpaired, but as it is a factor of the original number it must have a pair. Therefore the only pair possible is with itself, ie the number is this factor squared and the number is a perfect square.

for example: for 144, the factors are: {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144} which can be paired as: {1, 144}, {2, 72}, {3, 48}, {4, 36}, {6, 24}, {8, 18}, {9, 16} leaving the factor 12 unpaired, so it must pair with itself as {12, 12} which means 144 = 12² and is a perfect square (with an odd number of factors).