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Number Theory Level 3

3 37 m o d 111 = ? \large{3^{37}\mod111=?}

What's the remainder when 3 37 3^{37} is divided by 111 111 ?


The answer is 3.

Md Omur Faruque by MD Omur Faruque , 25, Bangladesh

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2 solutions

Md Omur Faruque
Aug 31, 2015

Note that, 3 37 m o d 111 = ( 3 × 3 36 ) m o d ( 3 × 37 ) 3^{37}\mod111=(3\times3^{36}) \mod(\color{#69047E} {3} \times37)

From Fermat's Little Theorem , we know that, a p − 1 ≡ 1 ( m o d p ) a^{p-1}\equiv1\pmod p , where p p is a prime number, a a is any arbitrary positive integer and a a & p p are coprime.

As, 37 37 is prime we get, 3 36 m o d 37 ≡ 1 ( m o d 37 ) 3^{36}\mod37\equiv \color{teal} {1} \pmod{37}

Thus, 3 37 m o d 111 = 3 × 1 = 3 3^{37}\mod111=\color{#69047E} {3} \times \color{teal} {1} =\color{#3D99F6} {\boxed {3}}

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Thats the kind of Brilliant solution!!!

Cleres Cupertino - 5 years, 9 months ago

I really loved the problem. At first it seemed beyond me as Euler's totient theorem and FLT does not work as 111 is divisible by 3 and 111 is not prime. I then realized the fact that 111 was 3*37. I love the fact where you made us use the prime factorization of 111 and this led to FLT.

Sathvik Acharya - 4 years, 2 months ago
Aaryan Maheshwari
Jun 28, 2019

Since by Fermat's Little Theorem, 3 37 ≡ 3 ( m o d 37 ) 3^{37}\equiv3\pmod{37} , automatically 3 37 ≡ 3 ( m o d 111 ) 3^{37}\equiv3\pmod{111} since 111 = 37 × 3 111=37\times3 .

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