Let us say, N = 3^{200} – 5^{100}
N = 9^{100} – 5^{100}.
This is a multiple of 9 - 5. Hence, 4, 2 and 1 are factors of the dividend.
N can also be written as 81^{50} – 25^{50}.
This is a multiple of 81 - 25 = 56. Hence, 56, 7, 8 are factors of the dividend.
Dividend = (81^{2})^{25} – (25^{2})^{25}
This is a multiple of 81^{2} - 25^{2}, 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? 3^{200} is a multiple of 3. 5^{100} 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.
Answer choice (a)
Correct Answer: 3.