List every factor of a number, along with its prime factors and how many factors it has.
Every number from 1 up to the square root is tested for division without remainder.
Each factor found gives a partner factor on the other side of the square root.
Prime factorisation repeatedly divides out the smallest prime that fits, until nothing is left.
Factor: A whole number that divides n with no remainder
Search range: Test only up to √n; factors come in pairs
Finding the factors of 360.
Number: 360
Divide out 2s: 360 = 2 × 2 × 2 × 45
Divide out 3s: 45 = 3 × 3 × 5
Prime factorisation: 2³ × 3² × 5
Factor count: (3+1) × (2+1) × (1+1) = 24
Answer: 360 has 24 factors; 2³ × 3² × 5
How can I tell how many factors a number has without listing them?
Take the prime factors, add 1 to each power, and multiply them together. For 360 = 2³ × 3² × 5¹, that gives 4 × 3 × 2 = 24 factors.
Why does a perfect square have an odd number of factors?
Factors normally come in pairs, giving an even total. But in a perfect square, the square root pairs with itself, so it is only counted once instead of twice, making the total an odd number.