Q35 / 108 · NTA Abhyas (JEE Main) 2020 · Mathematics · Binomial Theorem

The last two digits of the number ${(23)}^{14}$ are

A
$01$
B
$03$
Correct answer
C
$09$
D
$17$

Solution

{(23)}^{14} = {(529)}^{7} = {(530 - 1)}^{7}

= {{}^{7}C}_{0} {(530)}^{7} - {{}^{7}C}_{1} {(530)}^{6} + . \dots \dots . - {{}^{7}C}_{5} {(530)}^{2} + {{}^{7}C}_{6} (530) - 1

= 100 K + 7 \times 530 - 1

= 100 K + 3709

\Rightarrow

last two digits are

09

Hence, the correct answer is option: 'C'.

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