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If (65)x = (52)y, then what is the minimum value of x+y ?

Converting both sides to decimal,

6x+5 = 5y+2

We get, 5y - 6x = 3

Now, the first number has 5 and 6 as its digits. So x is definitely greater than 6. The second number has 5 and 2 as its digits. So, y is definitely greater than 5.

So, x can be 7,8,9,10..... and y can be 6,7,8,9.....

By inspection, if we take y as 9 and x as 7, then the equation is satisfied.

So, minimum value of (x+y) should be 16.

answered by Veteran (10.2k points) 6 43 108
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why  y as "9" not "6"?
Give it a try..
oh okay we are comparing x and y with (65)x =(52)y , I thought that we are just trying to find min value of (x + y).  thanks
Yes bro!!

Minimum value of  x+y is 16

answered by Junior (697 points) 1 5
we know that x>6 and y>5

6x+5=5y+2

=>   6x+3=5y

=>   3(2x+1)=5y

=>  2x+1 is divisible by 5 and is odd at the same time x>6

so minimum value possible for 2x+1 is 15

which gives x=7

=>from solving above equation we get y=9.

So minimum value of x+y is 7+9=16