Positive Integer Solutions

Number Theory Level 3

Find the number of positive integral solutions ( x , y ) (x, y) to the following equation:

2 x + 3 y = 763. 2x + 3y = 763.


The answer is 127.

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Viki Zeta by Viki Zeta , 19, India

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1 solution

Viki Zeta
Nov 12, 2016

2 x + 3 y = 763 2 x = 763 − 3 y x = 763 − 3 y 2 Now, x is a positive integer, ie, 1, 2, 3, 4, ... . 763 not is divisible by 2, so only if 2 divides ’763 - 3y’ x is an integer. 763 is odd, odd - odd = even, 2 divides even, therefore 3y is odd and 3y < 763 3 y < 763 y < 254.33 1 ≤ y < 254 Using AP there are 127 odd numbers from 1 to 254. ∴ No of solutions = 127 2x + 3y = 763 \\ 2x = 763 - 3y \\ x = \dfrac{763-3y}{2} \\ \text{Now, x is a positive integer, ie, 1, 2, 3, 4, ... . 763 not is divisible by 2, so only if } \\ \text{2 divides '763 - 3y' x is an integer. 763 is odd, odd - odd = even, 2 divides even, therefore 3y is odd and 3y < 763} \\ 3y < 763 \\ y < 254.33 \\ 1 \le y < 254\\ \text{Using AP there are 127 odd numbers from 1 to 254.} \\ \boxed{\therefore \text{No of solutions = } 127}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Good job! You can probably just stop at 2x = 763 - 3y and note that the RHS must be even by definition before jumping into the odd/even part.

Jason Dyer Staff - 4 years, 7 months ago

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hmm yeah.. Both apply same logic tho.

Viki Zeta - 4 years, 7 months ago

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