A number theory problem by abhishek alva

Number Theory Level 3

Find the last two digits of the decimal representation of 3 1999 + 2 1999 3^{1999}+2^{1999} .


The answer is 55.

Abhishek Alva by abhishek alva , 19, India

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1 solution

Ayush G Rai
Jan 1, 2017

Observe that 2 100 ≡ 76 ( m o d 100 ) 2^{100}\equiv 76 \pmod{100} and 3 100 ≡ 1 ( m o d 100 ) . 3^{100}\equiv 1 \pmod{100}. Thus,
2 1999 + 3 1999 ≡ 2 99 + 3 99 ≡ 88 + 67 ≡ 55 ( m o d 100 ) 2^{1999}+3^{1999}\equiv 2^{99}+3^{99}\equiv 88+67\equiv \boxed{55}\pmod{100}

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