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 rainbow a. the number 89 has dropdown with less than two, exactly two, more than two factors. the number 89 is checkbox dropdown options as above. (images of rainbow a, b, c, d are shown with different factor sets.)

Explanation:

First, determine the factors of 89:

89 is a prime number. A prime number is a number greater than 1 that has exactly two distinct positive divisors: 1 and itself. So the factors of 89 are 1 and 89.

Analyze each rainbow:

• Rainbow A: Shows factors 1, 3, 6, 13, 23, 89. But 3, 6, 13, 23 are not factors of 89 (since \( 89\div3\approx29.67 \), \( 89\div6\approx14.83 \), \( 89\div13\approx6.85 \), \( 89\div23\approx3.87 \)), so Rainbow A is incorrect.
• Rainbow B: Shows factors 1 and 89. This is correct because 1 and 89 are the only factors of 89.
• Rainbow C: Shows factors 1, 7, 13, 89. But \( 89\div7\approx12.71 \), \( 89\div13\approx6.85 \), so 7 and 13 are not factors of 89. Incorrect.
• Rainbow D: Shows factors 1, 5, 9, 17, 89. \( 89\div5 = 17.8 \), \( 89\div9\approx9.89 \), \( 89\div17\approx5.24 \), so these are not factors. Incorrect.

Now, answer the statements:

1. The factor rainbow that correctly shows the factors of 89 is Rainbow B (since only 1 and 89 are factors).
2. The number 89 has exactly two factors (1 and 89, as it's prime).
3. The number 89 is a prime number, which means it has exactly two distinct positive divisors, so it is "exactly two" (in terms of the number of factors) and it is a prime number (so the last statement, if it's about being prime, but based on the dropdown, since it has exactly two factors, the correct option for the number of factors is "exactly two", and the number 89 is a prime number, so the checkbox should be for "exactly two" in the factors and recognizing it's prime (but the dropdown for factors is "exactly two").

Correcting the first statement:

The initial selection of Rainbow A was wrong. The correct rainbow is Rainbow B.

For the number of factors:

Since 89 is prime, it has exactly two factors (1 and 89).

For the last statement (is 89...):

Since it has exactly two factors, it is a prime number, so the correct option is "exactly two" for the number of factors, and the number 89 is a prime number (so the checkbox should be for the statement about having exactly two factors, and recognizing it's prime, but based on the dropdown, the number of factors is "exactly two".

Final Answers:

• The factor rainbow that correctly shows the factors of 89 is \(\boldsymbol{\text{Rainbow B}}\).
• The number 89 has \(\boldsymbol{\text{exactly two}}\) factors.
• The number 89 is (a prime number, so the correct option for the checkbox is related to having exactly two factors, so the dropdown for factors is "exactly two", and the number 89 is a prime number, so the statement about the number of factors is "exactly two".)

(Note: The initial mistake was in the rainbow selection; the correct rainbow is B, not A. Then, since 89 is prime, it has exactly two factors.)

Answer:

First, determine the factors of 89:

89 is a prime number. A prime number is a number greater than 1 that has exactly two distinct positive divisors: 1 and itself. So the factors of 89 are 1 and 89.

Analyze each rainbow:

• Rainbow A: Shows factors 1, 3, 6, 13, 23, 89. But 3, 6, 13, 23 are not factors of 89 (since \( 89\div3\approx29.67 \), \( 89\div6\approx14.83 \), \( 89\div13\approx6.85 \), \( 89\div23\approx3.87 \)), so Rainbow A is incorrect.
• Rainbow B: Shows factors 1 and 89. This is correct because 1 and 89 are the only factors of 89.
• Rainbow C: Shows factors 1, 7, 13, 89. But \( 89\div7\approx12.71 \), \( 89\div13\approx6.85 \), so 7 and 13 are not factors of 89. Incorrect.
• Rainbow D: Shows factors 1, 5, 9, 17, 89. \( 89\div5 = 17.8 \), \( 89\div9\approx9.89 \), \( 89\div17\approx5.24 \), so these are not factors. Incorrect.

Now, answer the statements:

1. The factor rainbow that correctly shows the factors of 89 is Rainbow B (since only 1 and 89 are factors).
2. The number 89 has exactly two factors (1 and 89, as it's prime).
3. The number 89 is a prime number, which means it has exactly two distinct positive divisors, so it is "exactly two" (in terms of the number of factors) and it is a prime number (so the last statement, if it's about being prime, but based on the dropdown, since it has exactly two factors, the correct option for the number of factors is "exactly two", and the number 89 is a prime number, so the checkbox should be for "exactly two" in the factors and recognizing it's prime (but the dropdown for factors is "exactly two").

Correcting the first statement:

The initial selection of Rainbow A was wrong. The correct rainbow is Rainbow B.

For the number of factors:

Since 89 is prime, it has exactly two factors (1 and 89).

For the last statement (is 89...):

Since it has exactly two factors, it is a prime number, so the correct option is "exactly two" for the number of factors, and the number 89 is a prime number (so the checkbox should be for the statement about having exactly two factors, and recognizing it's prime, but based on the dropdown, the number of factors is "exactly two".

Final Answers:

• The factor rainbow that correctly shows the factors of 89 is \(\boldsymbol{\text{Rainbow B}}\).
• The number 89 has \(\boldsymbol{\text{exactly two}}\) factors.
• The number 89 is (a prime number, so the correct option for the checkbox is related to having exactly two factors, so the dropdown for factors is "exactly two", and the number 89 is a prime number, so the statement about the number of factors is "exactly two".)

(Note: The initial mistake was in the rainbow selection; the correct rainbow is B, not A. Then, since 89 is prime, it has exactly two factors.)