HCF with common remainders: Find the greatest number that divides 263, 935, and 1383 leaving a remainder of 7 in each case.

Introduction / Context:
When a number D divides several numbers leaving the same remainder r, D must divide the pairwise differences after subtracting r. This converts a “common remainder” problem into a Highest Common Factor (HCF) computation.

Given Data / Assumptions:

• Numbers: 263, 935, 1383.
• Common remainder r = 7.
• We seek the largest divisor D.

Concept / Approach:
If each number N leaves remainder 7 upon division by D, then D | (N − 7). So compute 263 − 7, 935 − 7, and 1383 − 7, and then take their HCF.

Step-by-Step Solution:

Verification / Alternative check:
Confirm all original numbers give remainder 7 when divided by 32: 263 = 32*8 + 7, 935 = 32*29 + 7, 1383 = 32*43 + 7.

Why Other Options Are Wrong:
31 and 35 do not divide all adjusted values (256, 928, 1376). 30 is not a factor of 256 or 928.

Common Pitfalls:
Forgetting to subtract the common remainder before finding the HCF; directly taking HCF of the raw numbers would be incorrect.

Final Answer:
32