Choose the largest number from the following that will perfectly divide $$(101)^5$$​ - 1.

Correct option is C

Solution:

​$$101^5 - 1 = (100 + 1)^5 - 1$$

$$(100 + 1)^5 = \binom{5}{0} \times (100)^0 + \binom{5}{1} \times (100)^1 + \binom{5}{2} \times (100)^2 + \binom{5}{3} \times (100)^3 + \binom{5}{4} \times (100)^4 + \binom{5}{5} \times (100)^5\\= 1 + 5 \times 100 + 5 \times \frac{4}{2} \times (100)^2 + 5 \times \frac{4}{2} \times (100)^3 + 5 \times (100)^4 + (100)^5\\(101)^5 - 1 = 1 + 5 \times 100 + 5 \times \frac{4}{2} \times (100)^2 + 5 \times \frac{4}{2} \times (100)^3 + 5 \times (100)^4 + (100)^5 - 1\\= 5 \times 100 \left[ 1 + 2 \times 100 + 2 \times 100^2 + 100^3 + 20 \times 100^3 \right]\\=500$$​​​