Factors of a positive integer typically occur in distinct pairs (a, b) such that a × b = N. The only time a factor pairs with itself is when a = b, meaning a² = N. Therefore, a number has an odd number of distinct factors if and only if it is a perfect square.

Factors of a positive integer typically occur in distinct pairs (a, b) such that a × b = N. The only time a factor pairs with itself is when a = b, meaning a² = N. Therefore, a number has an odd number of distinct factors if and only if it is a perfect square.

We test each of the four given options to determine which one is a perfect square. Testing each value: 72 is not a square (8² = 64, 9² = 81), 169 = 13² which is an exact integer square, 180 is not a square, and 200 is not a square.

We test each of the four given options to determine which one is a perfect square. Testing each value: 72 is not a square (8² = 64, 9² = 81), 169 = 13² which is an exact integer square, 180 is not a square, and 200 is not a square.

We can verify this directly using the formula for the number of divisors. The prime factorization of 169 is 13². The number of factors is 2 + 1 = 3, which are 1, 13, and 169. Since 3 is an odd number, 169 has an odd number of factors.

We can verify this directly using the formula for the number of divisors. The prime factorization of 169 is 13². The number of factors is 2 + 1 = 3, which are 1, 13, and 169. Since 3 is an odd number, 169 has an odd number of factors.

10. Which of the following has an odd number of factors? 72 169 180 200

Answer: $169$

Step-by-step solution

Step 1: Understand the Property of Numbers with Odd Factors

Factors of a positive integer typically occur in distinct pairs $(a, b)$ such that $a \times b = N$ . The only time a factor pairs with itself is when $a = b$ , meaning $a^2 = N$ . Therefore, a number has an odd number of distinct factors if and only if it is a perfect square.

Step 2: Check Each Given Option for Perfect Squares

We test each of the four given options to determine which one is a perfect square. Testing each value: $72$ is not a square ( $8^2 = 64$ , $9^2 = 81$ ), $169 = 13^2$ which is an exact integer square, $180$ is not a square, and $200$ is not a square.

Step 3: Verify by Prime Factorization

We can verify this directly using the formula for the number of divisors. The prime factorization of $169$ is $13^2$ . The number of factors is $2 + 1 = 3$ , which are $1$ , $13$ , and $169$ . Since $3$ is an odd number, $169$ has an odd number of factors.

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