HSPT Math Question of the Day

1. 6
2. 8
3. 12 (correct answer)
4. 15

Explanation: When you need to find the number of positive factors of a number, the most efficient approach is to use prime factorization. This method works because every factor of a number can be expressed as a product of its prime factors raised to various powers. First, let's find the prime factorization of 72. Start by dividing by the smallest prime: $72 = 8 \times 9 = 2^3 \times 3^2$. So $72 = 2^3 \times 3^2$. To find the total number of factors, use this formula: if a number has prime factorization $p_1^{a_1} \times p_2^{a_2} \times \ldots$, then the number of positive factors is $(a_1 + 1)(a_2 + 1)\ldots$ For $72 = 2^3 \times 3^2$, the number of factors is $(3 + 1)(2 + 1) = 4 \times 3 = 12$. This confirms answer choice C is correct. Let's examine why the other answers are wrong. Choice A (6) likely comes from incorrectly counting only some obvious factors like 1, 2, 3, 6, 8, 9, missing several others. Choice B (8) might result from counting the prime factorization elements incorrectly or forgetting the systematic approach. Choice D (15) could come from a calculation error in the formula or misidentifying the prime factorization. For factor problems on the HSPT, always use prime factorization rather than listing factors manually. The formula $(a_1 + 1)(a_2 + 1)\ldots$ is foolproof and prevents you from missing factors or making counting errors. Remember to add 1 to each exponent before multiplying.