Just amazing!

Number Theory Level 3

We denote { x } \{x\} as the fractional part of x x , given that

{ 3 1001 82 } = a b \large \left\{\frac{3^{1001}}{82}\right\} = \frac{a}{b}\

for coprime positive integers a a and b b , find the value of a + b a+b .


The answer is 85.

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U Z by U Z , 24, Guyana

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2 solutions

Sanjeet Raria
Oct 31, 2014

Since the remainder when 3 1001 = ( ( 3 8 ) 125 • 3 ) 3^{1001}=((3^8)^{125}•3) is divided by 82 82 is clealy 3 3 ⇒ 3 1001 = 82 k + 3 f o r s o m e n a t u r a l n u m b e r k \Rightarrow 3^{1001}=82k+3 \space for \space some \space natural \space number \space k ⇒ { 3 1001 82 } = { 82 k + 3 82 } \Rightarrow \{\frac{3^{1001}}{82}\}=\{\frac{82k+3}{82}\} { k + 3 82 } = 3 82 \{k+\frac{3}{82}\}= \frac{3}{82} Hence the answer is 3 + 82 = 85 3+82=\boxed{85}

17 Helpful 0 Interesting 0 Brilliant 0 Confused

Exactly , that's why its just amazing

U Z - 6 years, 7 months ago

How is it clearly 3 when divided by 82?

Tarush Govil - 5 years, 6 months ago
William Isoroku
Oct 10, 2015

Just do 3 1001 3^{1001} modulus 82 82 and get the remainder. Divide the remainder by 82

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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