Linear Diophantine equation
- Olympiad · IOQM
A linear Diophantine equation is an equation like 3x + 5y = 22 in which x and y must be integers.
It is named after Diophantus, a Greek mathematician. On its own, x + y = 5 has endless real answers. Asking for whole numbers only changes the problem. It becomes a question about divisibility.
The key rule: ax + by = c has integer answers exactly when the HCF of a and b divides c. Once you find one answer, all the others follow a simple pattern.
A small example
You have stamps worth 3 and 5. Can you make exactly 22? This asks for whole numbers with 3x + 5y = 22.
- The HCF of 3 and 5 is 1, which divides 22. So answers exist.
- Try y = 0, 1, 2. At y = 2, 3x = 22 - 10 = 12, so x = 4.
- So 4 stamps of 3 and 2 stamps of 5 make 22.
Every integer answer is x = 4 + 5t and y = 2 - 3t, for any integer t. With t = -1 you get x = -1, y = 5. Check: -3 + 25 = 22. Now try 4x + 6y = 9. The left side is always even, and 9 is odd. The HCF of 4 and 6 is 2, which does not divide 9. So there is no answer.
Where it fits
This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under number theory. The Mathematics Teachers' Association (India), MTA(I), conducts IOQM, the first stage of the Mathematical Olympiad Programme that HBCSE organises for NBHM.
Sources
Facts last checked against these sources on 30 September 2026.
- Syllabus for Mathematical Olympiad (Number Theory) · HBCSE, TIFR
- Mathematical Olympiad 2026-2027: stages of selection · HBCSE, TIFR
- Brochure: Mathematical Olympiads 2026-2027 · HBCSE, TIFR
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