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Number Theory Level 4

Find the remainder when 9 9 999999 99^{999999} is divided by 23.


The answer is 22.

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Akshat Sharda by Akshat Sharda , 21, India

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1 solution

Harsh Khatri
Feb 29, 2016

9 9 999999 ≡ ( 7 22 ) 45454 ⋅ 7 11 ≡ 1 45454 ⋅ ( 7 2 ) 5 ⋅ 7 1 ≡ ( 3 5 ) ⋅ 7 ≡ 13 ⋅ 7 ≡ 22 ( m o d 23 ) 99^{999999} \equiv \big(7^{22}\big)^{45454} \cdot 7^{11} \equiv 1^{45454} \cdot \big( 7^{2} \big)^{5} \cdot 7^1 \equiv (3^5) \cdot 7 \equiv 13\cdot 7 \equiv \boxed{22} \pmod{23}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice one line solution !! :-)

Akshat Sharda - 5 years, 3 months ago

It should be 7 ? how have you converted 9 9 999999 99^{999999} to the powers in 7 ??

Chirayu Bhardwaj - 5 years, 2 months ago

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