Find the greatest number that divides 43, 91 and 183 leaving the same remainder in each case.
The answer is 4. If a number leaves the same remainder for all three, it divides their differences exactly: 48, 92 and 140. HCF(48, 92, 140) = 4, and indeed each number leaves remainder 3 when divided by 4.
Explain it simply
Suppose three friends each have some sweets and, after packing them into boxes of the same size, each has the same number left over. Then the gap between any two friends' totals must fit exactly into whole boxes. So the box size divides every difference. The biggest such box size is the HCF of the differences.
Example
Question: Find the greatest number that divides 43, 91 and 183 leaving the same remainder.
- Differences: 91 - 43 = 48, 183 - 91 = 92, 183 - 43 = 140.
- HCF(48, 92): 92 = 48 x 1 + 44, 48 = 44 x 1 + 4, 44 = 4 x 11. So HCF = 4.
- HCF(4, 140) = 4.
- Answer: 4.
Check: 43 = 4 x 10 + 3, 91 = 4 x 22 + 3, 183 = 4 x 45 + 3. Same remainder 3 each time.
const g = (a, b) => (b === 0 ? a : g(b, a % b));
console.log(g(g(48, 92), 140)); // Output: 4
console.log([43, 91, 183].map(n => n % 4)); // Output: [3,3,3]What to say in the interview
"Same remainder means the divisor goes exactly into every difference, so I take the HCF of 48, 92 and 140, which is 4. Each number leaves 3, which confirms it."
Common mistakes
- Taking the HCF of the numbers themselves (which is 1 here).
- Using only two of the differences when the third would cut the HCF further. Two are enough mathematically, but check all three when in doubt.