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The largest value of n , for which $40^{\mathrm{n}}$ divides 60 !, is
[JEE MAIN 24 Jan 2026 S II]
(1) 13
(2) 11
(3) 12
(4) 14Solution
\[ 40=8 \times 5=2^{3} \times 5 \]Exponent of 2 in $60!=\left[\frac{60}{2}\right]+\left[\frac{60}{4}\right]+\left[\frac{60}{8}\right]+\left[\frac{60}{16}\right]+\left[\frac{60}{32}\right]+\left[\frac{60}{64}\right]+\cdots$
\[ =30+15+7+3+1+0+\cdots=56 \]Exponent of 5 in $60!=\left[\frac{60}{5}\right]+\left[\frac{60}{25}\right]+\left[\frac{60}{125}\right]+\cdots=12+2+0=14$
\[ 60!=2^{56} 5^{14}(\mathrm{~K})=2^{14}\left(2^{42}\right) 5^{14} \mathrm{~K}=2^{14}\left(2^{3} \times 5\right)^{14} \mathrm{~K}=2^{14}(40)^{14} \mathrm{~K} \]∴ Maximum value of n is 14
Answer (4)
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