Recall that a number p with exactly two factors (itself and 1) is called a prime number. (For the avoidance of doubt: 1 is NOT a prime number.)

The challenge this week is to investigate numbers with different numbers of factors.

1(i) Find (at least) four numbers with exactly three factors.

 (ii) Can you describe all numbers with exactly three factors?

2(i) Each of the numbers 6, 8, 10, 15, 21 have exactly four factors. But one is the odd one out. Which one?

(ii) Two different kinds of number have exactly four factors. Describe them.

 

3 Define d(n) = number of factors of n. So e.g. d(1) = 1, d(3) = 2, d(6)=4, d(20) = 6.

(i) What is d(600)?

(ii) Let p be a prime. What is d(pⁿ)?

(iiiLet p₁, p₂, p₃, ..., p_n be primes. What is d(p₁×p₂×p₃×...×p_n)?

(iv) Describe how to find d(n) in general by writing a number n as a product of powers of its distinct prime factors (e.g. 300 = 2²*3*5²).

(v) For what numbers n is d(n) = 12?