Agree or Disagree

Number Theory Level 2

Do you agree or disagree that 8 2016 − 3 2016 8^{2016}-3^{2016} is divisible by 5?

Disagree because it is false Agree because it is true

Hana Wehbi by Hana Wehbi , 51, USA

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2 solutions

展豪 張
May 29, 2016

Relevant wiki: Modular Arithmetic - Problem Solving - Basic

8 2016 − 3 2016 ≡ 3 2016 − 3 2016 = 0 ( mod 5 ) 8^{2016}-3^{2016}\equiv 3^{2016}-3^{2016}=0(\text{mod }5)

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice work.

Hana Wehbi - 5 years ago

8 2016 − 3 2016 = ( 8 − 3 ) ( 8 2015 + 3 ⋅ 8 2014 + . . . + 3 2014 ⋅ 8 + 3 2015 ) = 5 ( 8 2015 + 3 ⋅ 8 2014 + . . . + 3 2014 ⋅ 8 + 3 2015 ) 8^{2016}-3^{2016}=(8-3)({ 8 }^{ 2015 }+3\cdot { 8 }^{ 2014 }+...+{ 3 }^{ 2014 }\cdot 8+{ 3 }^{ 2015 })=5({ 8 }^{ 2015 }+3\cdot { 8 }^{ 2014 }+...+{ 3 }^{ 2014 }\cdot 8+{ 3 }^{ 2015 }) , clearly divisible by 5 5 .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Yes, it is true, you used the factorization of a n − b n a^{n}-b^{n} = ( a − b ) ( a n − 1 + a n − 2 b + . . . . . . + b n − 2 a + b n − 1 ) (a-b)(a^{n-1}+a^{n-2}b+......+b^{n-2}a+b^{n-1})

Hana Wehbi - 5 years ago

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