Find the largest value that divides 639 and 1468 that leaves remainder 4 and 3…

Concept: To find the largest number that divides two given numbers leaving specified remainders, first subtract each remainder from its own number to get two exact multiples of the required number. That required number is then the HCF (Highest Common Factor) of these two reduced values, found using the Euclidean algorithm.

Applying this to the given numbers:

1. Subtract the remainders from each number to get exact multiples: 639 − 4 = 635, and 1468 − 3 = 1465.

2. Divide the larger reduced value by the smaller: 1465 = 2 × 635 + 195.

3. Divide 635 by the remainder 195: 635 = 3 × 195 + 50.

4. Divide 195 by the remainder 50: 195 = 3 × 50 + 45.

5. Divide 50 by the remainder 45: 50 = 1 × 45 + 5.

6. Divide 45 by the remainder 5: 45 = 9 × 5 + 0 — the remainder becomes 0, so the algorithm stops here.

7. The last non-zero remainder is 5, so HCF(635, 1465) = 5, which is the required largest divisor.

Cross-check: 635 = 5 × 127 and 1465 = 5 × 293, and since 127 and 293 are both prime with no common factor, 5 is indeed their HCF. Verifying directly against the original numbers: 639 ÷ 5 gives quotient 127 and remainder 4, and 1468 ÷ 5 gives quotient 293 and remainder 3 — both match the conditions stated in the question.

So the largest value satisfying both conditions is 5.