Find the largest number which divides 1305, 4665 and 6905 leaving same…

Concept: When a single divisor leaves the same remainder on dividing several numbers, that divisor must exactly divide every pairwise difference between those numbers — the remainders cancel out in the difference. So the largest such divisor is the HCF of all the pairwise differences.

Working: Let the three numbers be x = 1305, y = 4665, z = 6905.

1. Pairwise difference: y − x = 4665 − 1305 = 3360

2. Pairwise difference: z − y = 6905 − 4665 = 2240

3. Pairwise difference: z − x = 6905 − 1305 = 5600

4. HCF(3360, 2240) via the Euclidean algorithm: 3360 = 1 × 2240 + 1120, then 2240 = 2 × 1120 + 0, so HCF(3360, 2240) = 1120

5. HCF(1120, 5600): 5600 = 5 × 1120 + 0, so the HCF of all three differences is 1120

6. Divide any one original number by 1120 to get the common remainder: 1305 = 1 × 1120 + 185, so the remainder is 185

Check: Verify against the other two numbers: 4665 = 4 × 1120 + 185, and 6905 = 6 × 1120 + 185 — all three numbers leave the same remainder, 185, on dividing by 1120, confirming both values.

Answer: The largest number is 1120 and the common remainder is 185.