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What is the unit digit in 189^723? A] 7 B] 9 C] 1 D] 3 Options : A B C D

Question

What is the unit digit in 189^723?

A] 7
B] 9
C] 1
D] 3

Options :
A
B
C
D

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Solution

1. Break Down the Problem

To find the unit digit of 189723 189^{723} , we only need to consider the unit digit of the base number, which is 9 9 in this case. Therefore, we will find the unit digit of 9723 9^{723} .

2. Relevant Concepts

The unit digits of the powers of 9 9 follow a pattern:

  • 91 9^1 has a unit digit of 9 9
  • 92 9^2 has a unit digit of 1 1
  • The pattern repeats every two powers: 9,1,9,1, 9, 1, 9, 1, \ldots

3. Analysis and Detail

To determine the unit digit of 9723 9^{723} , we find the exponent 723 723 modulo 2 2 : 723mod2=1 723 \mod 2 = 1 Since 723 723 is odd, the unit digit corresponds to the first element in the pattern which is 9 9 .

4. Verify and Summarize

Through our analysis, we see that the unit digit of 91 9^1 is 9 9 . Since 723 723 is odd, 9723 9^{723} also has a unit digit of 9 9 .

Final Answer

The unit digit in 189723 189^{723} is 9 (Option B).

This problem has been solved

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