# Polynomials

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## Factors

How many of the following are factors of 3200 – 5100?
1. 7
2. 16
3. 53
4. 12

1. 3
2. 2
3. 1
4. All of the above

Choice A. 3

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## Detailed Solution

Let us say, N = 3200 – 5100
N = 9100 – 5100.
This is a multiple of 9 - 5. Hence, 4, 2 and 1 are factors of the dividend.
N can also be written as 8150 – 2550.
This is a multiple of 81 - 25 = 56. Hence, 56, 7, 8 are factors of the dividend.
Dividend = (812)25 – (252)25
This is a multiple of 812 - 252, which is equal to (81 - 25) (81 + 25) = 56 × 106 = 7 × 8 × 2 × 53
Hence, 53 is also a factor of the dividend. The ‘2’ from this could combine with the 8 of the previous factor to mean that 16 is also a factor of the dividend.
The one exception of this list is ‘12’. Why? 3200 is a multiple of 3. 5100 is not a multiple of 3. If a non–multiple of 3 is subtracted from a multiple of 3, the result will be a non–multiple of 3.
Hence, 3 is not a factor of the dividend. => 12 is not a factor of this number.
So, three of the numbers are factors.

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