SSC CGL 2025 Tier-II · Paper-I, 19 January 2026 · Mathematical Abilities · Question 3 of 150

We transcribed this text from the pictures of a candidate's response sheet for this paper; the answer is the key SSC marked with a green tick on that sheet. Solution last reviewed 11 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. This paper reaches us as a candidate's response sheet saved as pictures, so we transcribed the text from those pictures, and a second reader checked every question against them; where the layout carries meaning, we show the picture too. Each answer is the key SSC marked with a green tick on that sheet. SSC released a tentative key for challenge and later its final one, both only to candidates behind a login, and the sheet does not say which of the two it carries. UnlockIAS wrote the solutions.

We doubt SSC's key on this question.

A direct count of 1 to 120 finds 25 numbers divisible by 6 or 8 but not by 24 (30 multiples of 6 or 8, less the 5 multiples of 24), so the probability is 25⁄120 — option (a) — not the 23⁄120 that SSC's key marks.

The answer below is still the option SSC's key marks, and we solve to that option.

A number is chosen at random from the set {1, 2, 3, ... , 120}. What is the probability that the number chosen is divisible by 6 or 8 but not divisible by 24?

(a)25⁄120
(b)19⁄120
(c)21⁄120
(d)23⁄120

Answer

Why

Correct — D. This card follows SSC's key, 23⁄120 at option (d). The count below reaches 25, not 23; the note on the key records the gap.

Multiples of 6 in 1 to 120: 120 ÷ 6 = 20
Multiples of 8: 120 ÷ 8 = 15
Multiples of both, i.e. of LCM 24: 120 ÷ 24 = 5

Divisible by 6 or 8: 20 + 15 − 5 = 30
Every multiple of 24 is among those 30, so remove all 5
Favourable: 30 − 5 = 25

Direct count: 25⁄120
SSC's key: 23⁄120 → option (d)

Why the others are wrong

(a)
25⁄120 — 25⁄120 is what the direct count gives: 30 numbers divisible by 6 or 8, less the 5 multiples of 24. SSC's key marks 23⁄120 at (d); the note on the key sets out the gap.
(b)
19⁄120 — 19 is not reached by either route. Multiples of 6 but not 24 number 20 − 5 = 15, multiples of 8 but not 24 number 15 − 5 = 10, and 15 + 10 = 25.
(c)
21⁄120 — 21 would mean removing nine numbers from the 30. The stem excludes the multiples of 24, and 1 to 120 holds five of them: 24, 48, 72, 96 and 120.

Concept

Counting "divisible by 6 or 8" uses inclusion–exclusion: add the two lists, then subtract the overlap once, because numbers divisible by both were counted twice.

The overlap is the multiples of the LCM, not the product. LCM(6, 8) = 24, so the overlap is the multiples of 24, not of 48.

Here "not divisible by 24" removes that overlap altogether. So the favourable numbers are those divisible by exactly one of 6 and 8.

Listing the favourable numbers settles the count at 25: 6, 8, 12, 16, 18, 30, 32, 36, 40, 42, 54, 56, 60, 64, 66, 78, 80, 84, 88, 90, 102, 104, 108, 112, 114.

SSC's key, the green tick on the response sheet, marks 23⁄120, and a second candidate's sheet carries the same key. The sheet does not say whether it holds the tentative key or the final one.

Key facts

n(A or B) = n(A) + n(B) − n(A and B).
A number divisible by both 6 and 8 is divisible by their LCM, 24.
From 1 to 120 there are 20 multiples of 6, 15 multiples of 8 and 5 multiples of 24.

Study next

Inclusion–exclusion with three divisorsCounting the multiples of k up to N as the whole part of N ÷ kProbability as favourable ÷ total for a number drawn at random from a set

Common traps

Taking the overlap as multiples of 48, the product, instead of 24, the LCM.
Stopping at 30⁄120. The union step subtracts the multiples of 24 once to stop double counting, and the "not divisible by 24" clause then removes them altogether.

Here the probability is a wrapper around a divisibility count. Finding the multiples of a number inside a range is also the first step of 12 Sep 2025, 09:00, Quant Q.7, which averages the integers between 100 and 250 exactly divisible by 11.

Related PYQs

No directly related past PYQ was found.

Topic: Statistics and probability — all 6 SSC CGL questions on it, every solved shift.