What is the highest power of 2 that can divide 50! (50 factorial)?

Concept: For a prime p and a positive integer n, the exponent of p in the prime factorization of n! (Legendre's Formula) equals the sum of floor(n / pk) for k = 1, 2, 3, ... until pk exceeds n. Each term counts how many multiples of pk lie between 1 and n, since every such multiple contributes an extra factor of p.

Application: here p = 2 and n = 50.

1. floor(50/2) = 25 - multiples of 2 up to 50.

2. floor(50/4) = 12 - multiples of 4, each contributing one extra factor of 2.

3. floor(50/8) = 6 - multiples of 8, each contributing yet another factor of 2.

4. floor(50/16) = 3 - multiples of 16.

5. floor(50/32) = 1 - multiples of 32.

6. floor(50/64) = 0 - since 64 is greater than 50, no further terms exist and the series terminates.

7. Sum: 25 + 12 + 6 + 3 + 1 = 47.

Cross-check via the digit-sum form of the same formula: v2(50!) = (50 - s)/(2 - 1), where s is the sum of 50's binary digits. Since 50 = 32 + 16 + 2 = 25 + 24 + 21, its binary representation 110010 has three 1s, so s = 3, giving v2(50!) = (50 - 3)/1 = 47 - the same result obtained above, and floor(50/64) = 0 further confirms the direct summation correctly terminated at 25 = 32.

Therefore, the highest power of 2 that divides 50! is 47.