Double Digits
What two digit numbers have the property that when the digits are reversed and the digits are then added to the new number, it equals the original number? For example, with the number 72:
72 =
27 =
27 + 2 + 7 =
36, which doesn't work.
Find the sum of all numbers that have this property.
The answer is 54.
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1 solution
Let the number be 10a + b,
such that a > b
the new number will be 10b + a
now, 10a + b - (10b + a) = 9(a - b)
.'.the difference between the original and the new number will always be an integer multiple of 9.
now, 10b+a+a+b = 10a+b
10b+a+a+b = 10b+a+9(a-b)
.'.a+b = 9(a-b)
.'.(a+b)/(a-b) = 9
a and b are single digit positive integers.
.'. a+ b = 9
.'.a- b = 1
.'.2a=10 => a=5, b=4
there is no other possibility so, 54