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I am new to stack exchange.I have a problem.For which I am unable to find an answer on the net.The question goes like this

"The sum of six different multiples of 3 is 66.

All the numbers are natural numbers.

The largest number among them is? "

Options:

a)21

b)30

c)27

d)24

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  • $\begingroup$ Have you made any attempts so far? $\endgroup$
    – TomGrubb
    Dec 19, 2016 at 1:41
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    $\begingroup$ Hint: compare to "6 different natural numbers add up to 22, then the largest can be $\cdots$". $\endgroup$
    – dxiv
    Dec 19, 2016 at 1:43

3 Answers 3

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Well, $3+6+9+12+15+18=3\cdot(1+2+3+4+5+6)=3\cdot21=63.$ What does that tell you, since we want them all to be distinct?

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You want the other numbers to be as small as possible, so pick 3,6,9,12,15 and subtract the sum from 66

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    $\begingroup$ That's a bit too informal in my opinion. Why do we want the other numbers to be as small as possible? If the sum was $99$, your solution would say the answer must be $54$... $\endgroup$
    – TMM
    Dec 19, 2016 at 12:51
  • $\begingroup$ Yeah, you're right, I guess I assumed the fact that no other configuration fits the question was self-explanatory... $\endgroup$
    – dasaphro
    Dec 19, 2016 at 20:55
  • $\begingroup$ Sure, if $3, 6, 9, 12, 15, x$ is not the unique solution, then there is no unique solution to the question, which is (presumably) a contradiction with the fact that this is a constructed exercise of some kind. But it's a small step to actually argue that $3, 6, 9, 12, 15, x$ is indeed the unique solution, without having to rely on the correctness of the question. It's better to show $P$ than to show $Q \implies P$ ;) $\endgroup$
    – TMM
    Dec 19, 2016 at 22:02
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Try dividing all of the numbers in this problem by $3$ (save the number of numbers in the sum). You can do this because they're all multiples of $3$, so you identify each number with how many $3$s it has. You can rephrase this problem as:

Six different natural numbers sum to $22$. The largest of them is: $7, 10, 9$, or $8$.

Now you can say that the five smallest numbers are at least $1,2,3,4,$ and $5$. What conclusion can you make now, if the six sum to $22$?

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