Since 15 is congruent to 1 modulo 7, 15^2010 is congruent to 1^2010 = 1 modulo 7 and 16 is congruent to 2 modulo 7, 16^2011 is congruent to 2^2011 modulo 7. Imagine that 2^3 is congruent to 1 modulo 7, then 2^2010 is congruent to 1^2010 = 1 modulo 7 and hence, 2^2011 is congruent to 2 modulo 7. Adding the remainders (since positive congruence), 15^2010 + 16^2011 is congruent to 1 + 2 = 3.

Looks like you have a modular arithmetic problem there - have a look at the 2 technique trainer articles Modular Arithmetic and Euler's Theorem first, and see how you go.
–
Ryan Carson
·
3 years, 9 months ago

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TopNewestSince 15 is congruent to 1 modulo 7, 15^2010 is congruent to 1^2010 = 1 modulo 7 and 16 is congruent to 2 modulo 7, 16^2011 is congruent to 2^2011 modulo 7. Imagine that 2^3 is congruent to 1 modulo 7, then 2^2010 is congruent to 1^2010 = 1 modulo 7 and hence, 2^2011 is congruent to 2 modulo 7. Adding the remainders (since positive congruence), 15^2010 + 16^2011 is congruent to 1 + 2 = 3.

Answer: 3 is the remainder. – John Ashley Capellan · 3 years, 9 months ago

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Do it by modulo function. – Baidehi Chattopadhyay · 3 years, 3 months ago

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Looks like you have a modular arithmetic problem there - have a look at the 2 technique trainer articles Modular Arithmetic and Euler's Theorem first, and see how you go. – Ryan Carson · 3 years, 9 months ago

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4 – Krishn Kumar Gupta · 3 years, 8 months ago

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