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Modulus of a number cannot be negative. A simple idea that somehow escapes our mind every now and then.

## Linear Equations - Integer Solutions

2x + 5y = 103. Find the number of pairs of positive integers x and y that satisfy this equation.
1. 9
2. 10
3. 12
4. 20

Choice B. 10

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## Detailed Solution

Rearranging the equation, we get:
2x = 103 – 5y

This says that when you subtract a multiple of 5 from 103, you get an even number. You have to subtract an odd multiple of 5 from 103 in order to get an even number. There are 20 multiples of 5 till 100, ten of which are odd. (Note that you cannot subtract 105, or higher multiples as they result in a negative value for x.)

So, y can have ten integer values. x also has 10 integer values, each corresponding to a particular value of y.

y = 1, gives us a potential value for x, so do y = 3, 5, 7….19. y can take 10 values totally.

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