Last two!

Number Theory Level 2

Find the last two digits of 2 1 43 \large 21^{43}

23 21 61 53 31

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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1 solution

2 1 43 ≡ 2 1 43 mod ϕ ( 100 ) (mod 100) Since gcd ⁡ ( 21 , 100 ) = 1 , Euler’s formula applies. ≡ 2 1 43 mod 40 (mod 100) Euler’s totient function ϕ ( 100 ) = 40 ≡ 2 1 3 (mod 100) ≡ ( 20 + 1 ) 3 (mod 100) ≡ ( 2 0 3 + 3 ( 2 0 2 ) + 3 ( 20 ) + 1 ) 3 (mod 100) ≡ 61 (mod 100) \begin{aligned} 21^{43} & \equiv 21^{\color{#3D99F6} 43 \text{ mod }\phi (100)} \text{ (mod 100)} & \small \color{#3D99F6} \text{Since }\gcd(21, 100) = 1 \text{, Euler's formula applies.} \\ & \equiv 21^{\color{#3D99F6} 43 \text{ mod }40} \text{ (mod 100)} & \small \color{#3D99F6} \text{Euler's totient function }\phi (100) = 40 \\ & \equiv 21^3 \text{ (mod 100)} \\ & \equiv (20+1)^3 \text{ (mod 100)} \\ & \equiv (20^3+3(20^2)+3(20)+1)^3 \text{ (mod 100)} \\ & \equiv \boxed{61} \text{ (mod 100)} \end{aligned}

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@Munem Sahariar , it is a standared in Brilliant.org that when numbers appear on their own, they don't need to be in LaTex. See the answer options. I am a moderator and I have edited them.

Chew-Seong Cheong - 3 years, 8 months ago

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Thanks....

Munem Shahriar - 3 years, 8 months ago

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