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Q46·CSAT · Prelims 2023

Cryptarithm pp + qq + rr = tt0

ReasoningCryptarithmeticCoding-decoding● Hard

Question

Let pp, qq and rr be 2-digit numbers where p < q < r. If pp + qq + rr = tt0, where tt0 is a 3-digit number ending with zero, consider the following statements:

1The number of possible values of p is 5.
2The number of possible values of q is 6.

Which of the above statements is/are correct?

Options

a

1 only

Answer
b

2 only

c

Both 1 and 2

d

Neither 1 nor 2

Explanation

Represent the identical-digit 2-digit numbers algebraically : pp = 10p + p = 11p qq = 11q, \quad rr = 11r

The sum equation becomes: 11p + 11q + 11r = tt0 \implies 11(p + q + r) = tt0

Since tt0 must be a multiple of 11 and end in 0, it must be a multiple of LCM(11, 10) = 110. Given p, q, r are single-digit numbers where p < q < r, the maximum possible sum is 7 + 8 + 9 = 24, meaning 11(p+q+r) \le 11 × 24 = 264. The only multiple of 110 in this range is 110 or 220.

Case 1: 11(p + q + r) = 110 \implies p + q + r = 10. Find single-digit combinations where p < q < r:

p = 1: pairs for (q, r) summing to 9 \rightarrow (2, 7), (3, 6), (4, 5).
p = 2: pairs for (q, r) summing to 8 \rightarrow (3, 5).

Case 2: 11(p + q + r) = 220 \implies p + q + r = 20.

p = 3: pairs summing to 17 \rightarrow (8, 9).
p = 4: pairs summing to 16 \rightarrow (7, 9).
p = 5: pairs summing to 15 \rightarrow (6, 9), (7, 8).

Let's compile the set of possible values for each digit:

For p: can be {1, 2, 3, 4, 5}. (Count = 5). Statement 1 is Correct.
For q: can be {2, 3, 4, 5, 6, 7, 8}. (Count = 7). Statement 2 is Incorrect.
For identical-digit sums, factor out 11 immediately and use the structural constraints of the placeholder answer (like ending in 0) to find the target multiple.

Answer: (a).

Question details

Year

2023

Paper

CSAT

Question

Q46

Section

Logical & Analytical Reasoning

Sub-topic

Cryptarithmetic

Type

Coding-decoding

Difficulty

Hard

Source hint

Cryptarithmetic puzzle

Same sub-topic — other years

Cryptarithmetic has appeared in multiple CSAT papers:

Q39

2025

Cryptarithm PQR - PS = PPT

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