Question

Find two numbers that each have exactly 16 factors, two of which are 8 and 12.​

Answer 1

the number is 96 it has exactly 16 factors hope it helps you

Answer 2

The two numbers we are looking for are 27,648 and 49,152.

Given: Numbers 16, 8, 12.

To find: Two numbers that each has exactly 16 factors.

Solution: Let's start by analyzing factors 8 and 12.

• Factors within 8 are 1, 2, 4, and 8. Therefore, 8 has a total of 4 determinants.
• Factors within 12 are 1, 2, 3, 4, 6, and 12. Therefore, 12 has a total of 6 factors.
• As we need to find two numbers that each have exactly 16 factors and two of these factors are 8 and 12, we can assume that these two numbers are multiples of 8 and 12, respectively.
• To have exactly 16 factors, both of these numbers need to be perfect squares because the total number of factors of a perfect square is always odd. Therefore, the two numbers we are looking for are of the form:

8^a * 12^b

• here a and b are positive integers.
• As we want the total number of factors to be 16, we need to find values of a and b that satisfy the equation:

(a + 1)(b + 1) = 16

We can solve this equation by trying out different pairs of values for a and b. Here are some possible pairs:

a = 1, b = 3: (1+1)(3+1) = 8 * 2 = 16

a = 3, b = 1: (3+1)(1+1) = 8 * 2 = 16

These pairs give us the numbers:

8^1 * 12^3 = 27,648

8^3 * 12^1 = 49,152

Both of these numbers have exactly 16 factors, two of which are 8 and 12.

Therefore, the two numbers we are looking for are 27,648 and 49,152.

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