What a large remainder!

Number Theory Level 2

If 9901^2 + 9907^2 + 9923^2 + 10007^2 is divided by 4. What is the Remainder? Please dont use calculator!!!!!!!


The answer is 0.

Sudhir Aripirala by Sudhir Aripirala , 21, India

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2 solutions

Rick B
Jan 7, 2015

990 1 2 + 990 7 2 + 992 3 2 + 1000 7 2 ≡ 1 2 + ( − 1 ) 2 + ( − 1 ) 2 + ( − 1 ) 2 9901^2+9907^2+9923^2+10007^2 \equiv 1^2+(-1)^2+(-1)^2+(-1)^2

= 1 + 1 + 1 + 1 = 4 ≡ 0 ( m o d 4 ) = 1+1+1+1 = 4 \equiv \boxed{0} \pmod{4}

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Nelson Mandela
Jan 6, 2015

so i took the last digits and added the last digit of their squares.

so, 1 + 49(9) + 9 + 49(9) = 1 + 9 + 9 + 9 = 28 which is divisible by 4. so The remainder must be 4. Moreover, 1 + 49 + 9 + 49 = 108 which is also divisible by 4. So answer is 0.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Actually I posted this problem of a theorem that i have noticed yesterday! Sum of squares of primes is congruent to number of primes taken if divide by 4@Nelson Mandela

Sudhir Aripirala - 6 years, 5 months ago

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Well,thanks. You are right.

Nelson Mandela - 6 years, 5 months ago

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