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# help

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The sum of the smallest two factors of an integer n is 6 and the sum of the two largest factor is 1122. Find n.

Oct 26, 2019

#1
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It could be 1,496:

1,496 =1 | 2 | 4 | 8 | 11 | 17 | 22 | 34 | 44 | 68 | 88 | 136 | 187 | 374 | 748 | 1496 (16 divisors)

Ignoring 1, then: 2 + 4 = 6 and 748 +374 =1,122

Oct 26, 2019
#2
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That wouldn't work since 1 and 1496 are also both factors. We know that the greatest factor of an integer n would n itself, and the smallest is 1. That would mean that the two smallest factors are 1 and 5, and the two greatest factors are n and 1122-n. We now use factor pairs to get that 1*n = 5(1122-n), or n = 5610-5n. Solving for n, we get that n = 935

Oct 27, 2019
#3
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The sum of the smallest two factors of an integer n is 6 and the sum of the two largest factor is 1122. Find n.

1 is a factor of every integer so the two smallest factors must be 1 and 5 which means that n is not divisable by 2 or by 3

The sum of the 2 largest factors is 1122.  So the number n must be smaller than 1122/2 = 561

So one is 561-x   and the other is 561+x

Maybe one is a multiple of 5           So maybe x=6t or 4t

x can't be 6t because 561 is divisable by 3 so 561*6t is also divisable by 3 which is not allowed.

that would give us      561+4t    and   561-4t

I cannot see any obvious reason why this can't be true.

I will try t values one at a time and see if I can find one that works

 t 1 565 557

That one was easy, t =1 is a contender.

Well 565 does not have 2 or 3 as a factor and neither does 557 so they could be right.

557+565 = 1122

factor(565) = 5*113

factor(557) = 1+557

557*565 = 314705

So the number could be 314705

At this point in tme I am not convinced that this is the only possible answer though.

Oct 27, 2019