SOLUTION: This year Ann’s age is the sum of the digits of her maths teacher’s age. In five years Ann’s age will be the product of the digits of her math teacher’s age at that time. H

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Question 1134437: This year Ann’s age is the sum of the digits of her maths teacher’s age. In five years Ann’s age will be the product of the digits of her math teacher’s age at that time. How old is Ann now?
Answer by greenestamps(13382) About Me  (Show Source):
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Let x and y be the tens and units digits of her teacher's age; that is, her teacher's age is 10x+y. Then

Ann's age is the sum of the digits of her teacher's age:
(1) A+=+x%2By

In 5 years, her age will be the product of the digits of her teacher's age then.

There are two possible cases:
(a) the units digit of her teacher's age is less than 5, so in 5 years the tens and units digits of his age will be x and y+5; or
(b) the units digit of his age is 5 or greater; in that case the tens and units digits of his age will be x+1 and y-5.

It turns out there is no solution for the first case. I will show how to solve the problem for the second case.

Ann's age in 5 years will be the product of the digits of her teacher's age then:
(2) A%2B5+=+%28x%2B1%29%28y-5%29

Subtracting (1) from (2) gives us

%28x%2B1%29%28y-5%29+-+%28x%2By%29+=+5
xy-5x%2By-5-x-y+=+5
xy-6x+=+10

This is a Diophantine equation -- one equation with two unknowns. It has either a single solution or a small number of solutions based on the fact that the variables are positive integers. Furthermore, in this problem, since the variables are the digits of the teacher's age, the variables have to have values that are positive single-digit integers.

The standard method for solving a Diophantine equation is to solve the equation for one variable in terms of the other and use the requirements that the variables have positive integer values to find the solution(s).

xy-6x+=+10
x%28y-6%29+=+10
x+=+10%2F%28y-6%29

x and y have to be positive single-digit integers; and 10 divided by (y-6) has to be a positive single-digit integer.

y=7 makes x=10/1=10, which is not allowed.
y=8 makes x=10/2=5.
y=9 makes x=10/3, which is not an integer.

So the solution should be with x=5 and y=8. Let's try it.

The teacher's age is 58; the sum of the digits of his age is 13; so Ann is 13.

In 5 years, the teacher will be 63; the product of the digits of his age is 18; Ann's age in 5 years will be 13+5=18.

SOLVED!