The last two digits of the number 23 14 are, 01, 03, 09, 17

23 14 = 529 7 = 530 - 1 7 = C 7 0 530 7 - C 7 1 530 6 + . … … . - C 7 5 530 2 + C 7 6 530 - 1 = 100 K + 7 × 530 - 1 = 100 K + 3709 ⇒ last two digits are 09

The last two digits of the number 23 14 are

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Question: Answered & Verified by Expert

The last two digits of the number

{(23)}^{14}

are

MathematicsBinomial TheoremJEE Main

Options:

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

Solution:

1067 Upvotes Verified Answer

The correct answer is:

09

{(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

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