SOLUTION: Find the sum of all the integers divisible by 7 between 32 and 5000. a) 1,514,285 b) 1,515,285 c) 1,615,185 d) 1,786,715 e) none of these

Algebra ->  Algebra  -> Divisibility and Prime Numbers -> SOLUTION: Find the sum of all the integers divisible by 7 between 32 and 5000. a) 1,514,285 b) 1,515,285 c) 1,615,185 d) 1,786,715 e) none of these      Log On

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Question 251848: Find the sum of all the integers divisible by 7 between 32 and 5000.
a) 1,514,285 b) 1,515,285 c) 1,615,185 d) 1,786,715 e) none of these
Found 2 solutions by richwmiller, jsmallt9:
Answer by richwmiller(9135) About Me  (Show Source):
You can put this solution on YOUR website!
this was solved yesterday

Answer by jsmallt9(3296) About Me  (Show Source):
You can put this solution on YOUR website!
Since the closest multiple of 7 to 5000 is 4998 (714*7), what we are looking for is:
35 + 42 + 49 + ... + 4992 + 4998
If we factor out 7 I think we'll see something we can figure out:
7(5 + 6 + 7 + ... + 713 + 714)
Inside the parentheses we see most of the numbers from 1 to 714. We are just missing 1, 2, 3 and 4. So if we can figure out this sum then we can multiply by 7 and have our answer.
The sum all the natural numbers up to n is given by the formula: S%5Bn%5D+=+%28n%28n%2B1%29%29%2F2. So we can find the sum of 1 + 2 + 3 + ... + 714 with: 714%2A715%2F2%29+=+255255. Now we need to "remove 1, 2, 3 and 4: 255255 - 1 - 2 - 3 - 4 = 255245. So 255245 is the sum of (5 + 6 + 7 + ... 714). Now we just multiply this by 7:
7*255245 = 1786715 which is answer (d)