CAT 2025 Slot 3 · Quant · Question 20

Write the number as \overline{abcd}. The conditions are a+b+c=15, b+c+d=16 and c=d+6.

Subtracting the first condition from the second, d-a=1, so d=a+1.

Combining with the third, c=d+6=a+7.

From the first condition, b=15-a-c=15-a-\left( a+7 \right)=8-2a.

Every digit must lie between 0 and 9, and a\ge 1 for a four-digit number.

The binding limit is c=a+7\le 9, which forces a\le 2; the condition b=8-2a\ge 0 would only allow up to a\le 4.

At a=1: b=6, c=8, d=2, giving 1682.

At a=2: b=4, c=9, d=3, giving 2493.

The difference is 2493-1682=811.

Shortcut. Expressing every digit in terms of a turns the question into finding the allowed range of a single variable. Only two values survive, so the two numbers appear immediately and no searching is required.