The sum of six different multiples of 3 is 66.All the numbers are natural numbers.The largest number among them is?

Question

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

Answer 1

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?

Answer 2

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

Answer 3

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 $22$?