Permutations and combinations (1 Viewer)

[fire and ice]

Hey guys,

Can you please help me with this question

By considering its prime factorisation, find the number of positive divisors of 315000

The answer is 120

 

[braintic]

Take a smaller number first, say 72 which is 2^3 × 3^2. First count the factors by listing them, as a check: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 26, 72 - there are 12 of them. How to get them systematically from the prime factorisation?
We have 3 factors of 2 and 2 factors of 3 at our disposal. So the factors of 72 can contain between 0 and 3 2s (4 options) and between 0 and 2 3s (3 options). 4 × 3 = 12 factors, as expected.

Now the prime factorisation of 315000 is 2^3 × 3^2 × 5^4 × 7^1. Using the same logic as before, number of factors = 4 × 3 × 5 × 2 = 120

 

[fire and ice]

thanks