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Can anyone help me with this?

If 738 consecutive integers are added together, where the 178th number in the sequence is 4,256,815, what is the remainder when this sum is divided by 6?

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    738 is divisible by 6 ($738= 123\cdot 6$), therefore you can pair up modulus 6 so the remainder will be 0. – 2017-02-26
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    For the upvoters: nope I was wrong. See Michael's answer. – 2017-02-26
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    If the 178 term is something something specific you know every term in the sequence. There's a basic formula for adding consecutive terms (or you can just multiply the average bu 738). Then just divide by 6. – 2017-02-27
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    This is not really a number theory problem, this is a "can I break down a big problem into small problems" problem. – 2017-02-27

1 Answers 1

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Let the integers be $x, x+1, \cdots, 4256815, \cdots, x+737$. Note that the sum of all of them is equal to $738x+(1+\cdots+737)$. $738$ is divisible by $6$, so the first term is always $0\pmod6$. In addition, we have $1+\cdots+737=\frac{737\cdot738}{2}=737\cdot369$. Note that $369$ is divisible by $3$; however, the product is odd, so it must be $0\pmod3$. Therefore, the sum must be $3\pmod6$, so the remainder is $3$.

P.S. The information that the $178$th number is $4256815$ is extraneous, you do not need it to solve the problem.