CSAT Solved Papers/ 2025/Q80

2025 CSAT — Q80

Quant Number theory 2.5 marks Medium

The difference between any two natural numbers is 1010. What can be said about the natural numbers which are divisible by 55 and lie between these two numbers?

  1. A There is only one such number.
  2. B There are only two such numbers.
  3. C There can be more than one such number. Answer
  4. D No such number exists.

Worked rationale

Let the two numbers be aa and a+10a + 10. Count the multiples of 55 strictly between them — i.e. in the open interval (a, a+10)(a,\ a+10), whose length is 1010. Such a window holds either one or two multiples of 55, depending on where aa falls:

  • a=1a = 1: interval (1,11)(1, 11) contains 55 and 1010 — two multiples.
  • a=5a = 5: interval (5,15)(5, 15) contains 1010 only (55 and 1515 are the excluded endpoints) — one multiple.

So the count is not fixed — it can be one, and it can be more than one. The only statement true in general is that there can be more than one such number.

Answer: (c) There can be more than one such number.

Why the other options miss

  • A
    missed a case: tests a case like a=5a=5 giving one multiple and generalises, missing a=1a=1 which gives two.
  • B
    missed a case: the mirror error — tests a=1a=1 (two multiples) and fixes on two, missing the one-multiple case.
  • D
    solved the wrong question: confuses “between” (open interval) with “endpoints excluded entirely,” wrongly concluding none lie inside.

Specialist insight

The key is that an interval of length 1010 does not pin the count of multiples of 55 — a gap of 1010 spans either one or two multiples of 55 depending on alignment, since 10/5=210 / 5 = 2 multiples’ worth but boundary effects drop it to one when an endpoint is itself a multiple. The disciplined move is to test two alignments (a=1a=1 and a=5a=5) and observe the count change; that immediately rejects the rigid “only one”/“only two” decoys and confirms the general “can be more than one.” Whenever a count depends on placement, the safe answer is the one phrased as a possibility, not a fixed number.

The trap, in one line

A length-1010 gap holds one or two multiples of 55 depending on alignment (a=5→a=5 \to one, a=1→a=1 \to two), so only "can be more than one" is true in general.

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