Positive integer coordinates

Number Theory Level 2

What is the number of positive integer solutions of 6 x + 9 y = 1400 6 x + 9 y = 1400 ?

0 1 3 2

Ossama Ismail by Ossama Ismail , 63, Egypt

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1 solution

For integer x x and y y , the LHS = 6 x + 9 y \text{LHS}= 6x+9y is divisible by 3, but the RHS = 1400 \text{RHS}=1400 is not divisible by 3. Therefore there is 0 \boxed 0 solution.

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In general the equation a x + b y = c ax + by = c has no integer solutions if c c does not divide into gcd ⁡ ( a , b ) \gcd(a,b) .

Ossama Ismail - 2 years, 3 months ago

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