Expert: Ahmed Salami - 8/13/2010

Question
QUESTION: If a number is divided by 46 the remainder is 7 and if the same number is divided by 45 the remainder is 8 what is the number

ANSWER: Hi Dhananjay,
Let the number be Z.
If y is the divisor when divided by 46, then
Z = 46y + 7
If x is the divisor when divided by 45, then
Z = 45x + 8
So,
46y + 7 = 45x + 8
46y = 45x + 1
This shows that there is an infinite set of numbers satisfying the condition. For the sake of simplicity, if we assume that all of x, y and Z are integers then its easy to see that x = y = 1 is a solution.
Therefore,
Z = 46(1) + 7 = 53

Regards


---------- FOLLOW-UP ----------

QUESTION: First of all thanks a lot for your accurate reply ,
But can you give me at least two pairs of integers taht will satisfy above equations or is there any thumb rule to find out x and y values in such cases

Answer
Hi Dhananjay,
Well, if we consider the line
46y = 45x + 1
y = (45/46)x + (1/46)
It has a slope of 45/46 and what this means is that any change in x results in a change (45/46) times that in y. Say, for instance, that x increases by 46, then y should increase by 45.
Starting from x = y = 1 (which we already know to be integer solutions), it follows that x = 1 + 46 = 47 and y = 1 + 45 = 46 would also be integer solutions.
We can then generally see that integer solutions would be of the form;
x = 1 + 46n
and
y = 1 + 45n
where n is an integer.
These results in a form of
Z = 53 + (45)(46)n

Regards