Find the greatest number by which 33, 278, and 285 can be divided so that the…

Concept:

If a number divides three integers and leaves the same remainder each time, it must divide every pairwise difference between those integers exactly, because the common remainder cancels out in subtraction. So the greatest such number is the HCF (highest common factor) of the pairwise differences.

Application:

1. Find the pairwise differences: 278 minus 33 equals 245, and 285 minus 278 equals 7.

2. Also, 285 minus 33 equals 252; use any two of these three differences to fix the HCF, keeping the third as a check.

3. Factor each difference into primes: 245 = 5 × 72, and 252 = 22 × 32 × 7. The only common prime factor is 7, so the HCF of 245 and 252 is 7.

4. Confirm against the reserved third difference: 7 divided by 7 is exactly 1, so 7 divides that difference exactly as well, confirming 7 is indeed the greatest common factor of all three pairwise differences.

5. Therefore, the greatest number satisfying the condition is 7.

Cross-check:

Dividing each original number by 7 confirms a single common remainder: 33 = 4 × 7 + 5, 278 = 39 × 7 + 5, and 285 = 40 × 7 + 5, a remainder of 5 in every case. This verifies that 7 works and cannot be improved on, since no larger common factor of the differences exists.