Question

Given that 2^A×3^B×5^13=20^D×18^12, where A,B, and D are postive integers, compute A+B+D.

Answer 1

Answer: 75

Explanation:

`2^A3^B5^(13)=20^D18^(12)`

⇒
`2^A3^B5^(13)=(2^2\cdot5^1)^D(2^1\cdot3^2)^(12)`

⇒
`2^A3^B5^(13)=(2^(2D)\cdot5^D)(2^(12)\cdot3^(24))`

⇒
`2^A3^B5^(13)=2^(2D+12)\cdot3^(24)\cdot5^D`

Now compare the like bases:

`2^A=2^(2D+12)` ⇒ A = 2D + 12

`3^B=3^(24)` ⇒ B = 24

`5^(13)=5^D` ⇒ D = 13

Next, let's solve for A:

A = 2D + 12

= 2(13) + 12

= 26 + 12

= 38

LAST STEP: Find the sum of A, B, and D

S = A + B + D

= 38 + 24 + 13

= 75