Question

considering only positive integer factors, which of the following integers has an even number of distinct factors? a 16 b 24 c 25 d 36

Explanation:

Step1: Recall factor properties

Numbers that are perfect squares have an odd number of distinct positive factors (since one factor is repeated, e.g., \( n = a^2 \), the factor \( a \) is counted once). Non - perfect squares have an even number of distinct positive factors (factors come in pairs \( (d, \frac{n}{d}) \)).

Step2: Analyze each option

• Option A: 16

\( 16=4^2 = 2^4 \). The factors of 16 are \( 1,2,4,8,16 \). The number of factors is \( 4 + 1=5 \) (using the formula for number of factors: if \( n=p^k \), number of factors is \( k + 1 \); if \( n=p_1^{k_1}p_2^{k_2}\cdots p_m^{k_m} \), number of factors is \( (k_1 + 1)(k_2 + 1)\cdots(k_m+1) \)). 5 is odd.

• Option B: 24

\( 24 = 2^3\times3^1 \). The number of factors is \( (3 + 1)\times(1+1)=4\times2 = 8 \). 8 is even. Let's verify by listing: factors of 24 are \( 1,2,3,4,6,8,12,24 \). There are 8 factors, which is even.

• Option C: 25

\( 25 = 5^2 \). The number of factors is \( 2 + 1=3 \) (factors: \( 1,5,25 \)). 3 is odd.

• Option D: 36

\( 36=6^2 = 2^2\times3^2 \). The number of factors is \( (2 + 1)\times(2 + 1)=9 \) (factors: \( 1,2,3,4,6,9,12,18,36 \)). 9 is odd.

Answer:

B. 24