The largest value of n, for which 40n divides 60! is [2026]

Question

The largest value of $n$ , for which $40^{n}$ divides $60!,$ is [2026]

Smart Achievers · Accepted Answer

(2)

$40^{n} = 2^{3 n} \times 5^{n}$

$E_{2} (60!) = [\frac{60}{2}] + [\frac{60}{2^{2}}] + [\frac{60}{2^{3}}] + [\frac{60}{2^{4}}] + [\frac{60}{2^{5}}]$

$= 30 + 15 + 7 + 3 + 1 = 56$

$E_{5} (60!) = [\frac{60}{5}] + [\frac{60}{5^{2}}]$

$= 12 + 2 = 14$

$40^{n} = {(2^{3})}^{n} \times 5^{n} = {(2^{3} \times 5)}^{n}$

$60! = 2^{56} \times 5^{14} . . . = 2^{14} \cdot {(2^{3} \cdot 5)}^{14}$

$∴$ $Maximum value of n is 14 .$

Want Us to Email you About Special Offers & Updates?