Maths Olympiad Prep

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Number theory Difficulty 4.9 AIME Find the answer

I have written a strictly increasing sequence of six positive integers, such that each number (besides the first) is a multiple of the one before it, and the sum of all six numbers is 79 . What is the largest number in my sequence?

A number or a short expression. Spacing and $ signs are ignored.

Solution

If the fourth number is \geq 12, then the last three numbers must sum to at least 12+12+ 212+2212=84>792 \cdot 12+2^{2} \cdot 12=84>79. This is impossible, so the fourth number must be less than 12. Then the only way we can have the required divisibilities among the first four numbers is if they are 1,2,4,81,2,4,8. So the last two numbers now sum to 7915=6479-15=64. If we call these numbers 8a,8ab(a,b>1)8 a, 8 a b(a, b>1) then we get a(1+b)=a+ab=8a(1+b)=a+a b=8, which forces a=2,b=3a=2, b=3. So the last two numbers are 16,48.

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