Bangladesh National Mathematical Olympiad 2016 Higher Secondary

[u][b]Problem 2:[/b][/u]
(a) How many positive integer factors does $6000$ have?

(b) How many positive integer factors of $6000$ are not perfect squares?
### Part (a)
1. **Prime Factorization of 6000:**
   \[
   6000 = 2^4 \times 3^1 \times 5^3
   \]
2. **Number of Positive Integer Factors:**
   The number of positive integer factors of a number \( n = p_1^{e_1} \times p_2^{e_2} \times \cdots \times p_k^{e_k} \) is given by:
   \[
   (e_1 + 1)(e_2 + 1) \cdots (e_k + 1)
   \]
   For \( 6000 = 2^4 \times 3^1 \times 5^3 \):
   \[
   (4 + 1)(1 + 1)(3 + 1) = 5 \times 2 \times 4 = 40
   \]

### Part (b)
1. **Perfect Square Factors:**
   A factor of \( 6000 \) is a perfect square if and only if all the exponents in its prime factorization are even. 
   \[
   6000 = 2^4 \times 3^1 \times 5^3
   \]
   To form a perfect square, the exponents must be even:
   - For \( 2 \), the exponent can be \( 0, 2, \) or \( 4 \) (3 choices).
   - For \( 3 \), the exponent can be \( 0 \) (1 choice).
   - For \( 5 \), the exponent can be \( 0 \) or \( 2 \) (2 choices).

   Therefore, the number of perfect square factors is:
   \[
   3 \times 1 \times 2 = 6
   \]

2. **Non-Perfect Square Factors:**
   The number of positive integer factors of \( 6000 \) that are not perfect squares is:
   \[
   40 - 6 = 34
   \]

The final answer is \( \boxed{34} \)