Factor world

Number Theory Level 3

If $N = 69^{5} + 5 × 69^{4} + 10 × 69^{3} + 10 × 69^{2} + 5 × 69 + 1$ , then the total number of factors of $N$ , is

125 216 250 200

2 solutions

X X
May 15, 2018

$N=70^5=2^55^57^5$ ,so $N$ has $6\times6\times6=216$ factors

Prem Chebrolu
Sep 13, 2018

This problem can be solved by using Euler's triangle! You notice the pattern, so this will be $(x+1)^{5}$ So, here $x = 69$ .

$\therefore$ , the number $N$ will be $(69+1)^{5}$

Which is, $70^{5}$ $=$ $2^{5}\cdot5^{5}\cdot7^{5}$

$\therefore$ , $N$ has, $6 \times {6} \times {6}$ $= 216$ factors!.

You can refer to this to know Euler's triangle