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  4. If N=(6+√34)7, then digit at unit's place of (N(1- N )10,( where . denotes fractional part function ), is equal to

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Q. If $N=(6+\sqrt{34})^{7}$, then digit at unit's place of $(N(1-$ $\{N\})^{10},($ where $\{.\}$ denotes fractional part function $)$, is equal to

Binomial Theorem

Solution:

$N=(6+\sqrt{34})^{7}=I+f$
$(6-\sqrt{34})^{7}=f'$
$f+f'=1$
$f'=1-f $
$N(1-\{N\})=N(1-f)$
$=(6+\sqrt{34})^{7}(6-\sqrt{34})^{7}=2^{7}$
$2^{70}$ unit's digit $=4$