MathDB

Let

a

and

b

be two-digit positive integers. Find the greatest possible value of

a+b

, given that the greatest common factor of

a

and

b

6

.
Proposed by Jacob Xu
Solution.

\boxed{186}

We can write our two numbers as

6x

and

6y

. Notice that

x

and

y

must be relatively prime. Since

6x

and

6y

are two digit numbers, we just need to check values of

x

and

y

from

2

through

16

such that

x

and

y

are relatively prime. We maximize the sum when

x = 15

and

y = 16

, since consecutive numbers are always relatively prime. So the sum is

6 \cdot (15+16) = \boxed{186}

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