Not a Polite Question
When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...
Problem
When asked how old she was, the teacher replied:
My age in years is not prime but odd, and when reversed and added to my age, you have a perfect square.
Or you can reverse and subtract, and again you have a perfect square.
Student Solutions
If her age is 10 t + u (where t is the tens digit and u is the units digit) then reversing the digits gives 10 u + t. and the sum is 11 t + 11 u, which is a multiple of 11. We know this has to be a square number. Ong Xing Cong from Singapore sent in the following solution.
11 t + 11 u = 11 x 11 = 121
t + u = 11
65 - 56 = 3 x 3
She is 65 years old.
The best solutions do not need trial and improvement methods and they show that the answer or answers found are the only possible answers. Knowing the digits add up to 11 ( t + u = 11), you can also use (10 t + u ) - (10 u + t ) = 9 t - 9 u = 9( t - u ) As this is also a square number you know ( t - u ) is either 1, 4, or 9. The solutions for 4 and 9 don't give whole number values for t in 0
t
9.