CSAT Solved Papers/ 2023/Q76

2023 CSAT — Q76

Quant Number theory 2.5 marks Hard

$AB$ and $CD$ are $2$ -digit numbers. Multiplying $AB$ with $CD$ results in a $3$ -digit number $DEF$ . Adding $DEF$ to another $3$ -digit number $GHI$ results in $975$ . Further $A, B, C, D, E, F, G, H, I$ are distinct digits. If $E = 0, F = 8$ , then what is $A + B + C$ equal to?

A 6 Answer
B 7
C 8
D 9

Worked rationale

With $E = 0, F = 8$ , the product is $DEF = \overline{D08}$ (a number of the form $100D + 8$ ), and $\overline{DEF} + \overline{GHI} = 975$ . The hundreds digit of $DEF$ is the same symbol $D$ that is the units digit of $CD$ , so the search is tightly linked.

Hunt for a $2$ -digit $\times$ $2$ -digit product of the form $\overline{D08}$ whose factor $CD$ ends in $D$ :

12 \times 34 = 408.

Here $CD = 34$ ( $C = 3, D = 4$ ), so $DEF = 408$ ( $D = 4, E = 0, F = 8$ ) — consistent. Then $AB = 12$ ( $A = 1, B = 2$ ), and

GHI = 975 - 408 = 567 \Rightarrow G = 5, H = 6, I = 7.

Check distinctness: $A,B,C,D,E,F,G,H,I = 1,2,3,4,0,8,5,6,7$ — all distinct (only $9$ unused). ✓

Therefore $A + B + C = 1 + 2 + 3 = 6$ .

Answer: (a) 6.

Why the other options miss

B

missed a case: lands on the rival factorisation $17 \times 24 = 408$ , but there $GHI = 567$ collides ( $I = 7$ with $B = 7$ ); failing the distinctness check should rule it out, not adjust the sum to $7$ .
C

an arithmetic slip: mis-factors $\overline{D08}$ or mis-adds $A+B+C$ after a wrong $AB$ .
D

solved the wrong question: ignores the $D$ -links- $CD$ -and- $DEF$ constraint and picks an arbitrary product ending in $8$ , inflating the digit sum.

Specialist insight

Two interlocking constraints prune the search to one answer: (1) $DEF = \overline{D08}$ forces a product ending in $08$ , and (2) the shared symbol $D$ ties $CD$ ‘s units digit to $DEF$ ‘s hundreds digit. Only $12 \times 34 = 408$ satisfies both and keeps all nine digits distinct after computing $GHI = 567$ . The near-miss $17 \times 24 = 408$ is the engineered trap — same product, but its $GHI$ repeats a digit. Always finish the distinctness check across all nine letters before locking the answer.

The trap, in one line

$DEF = \overline{D08}$ with $CD$ ending in $D$ : $12\times 34 = 408$ , $GHI = 567$ , all digits distinct $\Rightarrow A+B+C = 6$ (the $17\times 24$ branch fails distinctness) $=$ (a).

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