Question

four students tried to create a factor rainbow for the number 89. complete the statements below about the number 89. the factor rainbow that correctly shows the factors of 89 is dropdown with rainbow a, rainbow b, rainbow c, rainbow d. (images of rainbow a, b, c, d are shown on the right.)

Explanation:

Step1: Determine if 89 is prime

A prime number has only two factors: 1 and itself. Let's check if 89 is divisible by numbers other than 1 and 89. We test divisibility by primes less than \(\sqrt{89}\) (since \(\sqrt{89}\approx9.43\), we check primes 2, 3, 5, 7).

• 89 is odd, so not divisible by 2.
• \(8 + 9 = 17\), not divisible by 3, so 89 not divisible by 3.
• Ends with 9, not 0 or 5, so not divisible by 5.
• \(89\div7\approx12.71\), not an integer. So 89 is prime, factors are 1 and 89.

Step2: Analyze each rainbow

• Rainbow A: Has factors 3, 6, 13, 23 – these don't divide 89, so incorrect.
• Rainbow B: Shows 1 and 89 – correct, as 89 is prime.
• Rainbow C: Has 7, 13 – 7 and 13 don't divide 89, incorrect.
• Rainbow D: Has 3, 9, 17 – these don't divide 89, incorrect.

Answer:

Rainbow B