Question 722277: The product of a two-digit number and the same number with its digits reversed is 3154. Find the sum of the two numbers.
Found 2 solutions by ankor@dixie-net.com, DrBeeee:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! The product of a two-digit number and the same number with its digits reversed is 3154.
Find the sum of the two numbers.
We only have one equation and two unknown
:
let a = the 10's of the original equation
let b = the units
:
the sum:
s = (10a + b)+ (10b + a)
s = 11a + 11b
s = assume the sum will be a multiple of 11,
:
see if that works:
Let n = one of the numbers, we know that the difference between numbers that are reversed are multiple of 9, after some hit and miss came up with an equation with an integer solution:
:
n(n+45) = 3154
n^2 + 45n - 3154 = 0
solve for n
n = 38, then 83 is the other number
Check
38 * 83 = 3154
and
38 + 83 = 121 is the sum
Answer by DrBeeee(684)  (Show Source):
You can put this solution on YOUR website! I know the answer is 38*83 = 3154
So the sum is 38+83 = 121
However I've been unable to derive it. Sorry.
I've got
(1) n1 = 10x + y and
(2) n2 = 10y + x
Therefore the sum is
(3) n1 + n2 = 11x + 11y or
(4) sum = 11(x+y)
But I can't come up with 3 and 8 for x and y.
Nor can I solve
(5) n1*n2 = 3154
We know that n1 (or n2) has to be greater than 32 in order to get a second two digit number for the product.
Maybe we can say both are greater than 32 which means x or y has to be 3, then try 33*33, 34*43, 35*53, 36*63, 37*73, 38*83, 39*93. Indeed
the answer appear, but mathematician can't accept trial and error. There's got to be a better way.
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