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Finding Remainder of Numbers of the Form a^b

The last two ...

Question

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

A

01

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B

03

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C

09

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D

05

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Solution

The correct option is C $09$
$(23)^{14} = (529)^{7} = (530 - 1)^{7} =^{7} C_{0} (530)^{7} -^{7} C_{1} (530)^{6} + \dots +^{7} C_{6} (530)^{1} - 1 =^{7} C_{0} (530)^{7} -^{7} C_{1} (530)^{6} + . . . + 3710 - 1 = 100 k + 3709, k \in Z$

Therefore, last two digits are $09$ .

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Finding Remainder of Numbers of the Form a^b

Standard XII Mathematics

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