If N = 23 · 34 and M = 22 · 3 · 5, then find the number of factors of N that…

Concept: A positive integer is a common factor of two numbers exactly when it divides their greatest common divisor (gcd), so the number of factors shared by N and M equals the number of divisors of gcd(N, M).

Two supporting rules: build the gcd by keeping each prime to its minimum power across the two numbers (a prime missing from one number drops out), and count the divisors of a number pa · qb · … as the product (a + 1)(b + 1)… of one-more-than-each-power.

Application:

1. Write the prime factorisations: N = 23 · 34 and M = 22 · 31 · 51.

2. Take the minimum power of each shared prime — for 2: min(3, 2) = 2; for 3: min(4, 1) = 1. The prime 5 is absent from N, so it contributes nothing.

3. Assemble the gcd: gcd(N, M) = 22 · 31 = 12.

4. Count its divisors: (2 + 1)(1 + 1) = 3 · 2 = 6.

Cross-check: Listing the divisors of 12 directly gives 1, 2, 3, 4, 6, 12 — exactly 6 numbers, and each divides both 648 (= N) and 60 (= M). This confirms the count.

Result: 6 factors are common to N and M.