Integers not divisible by 2, 5, or 7

CAT 2020 Slot 3 · QA · Hard · Number theory

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How many of the integers $1, 2, \dots, 120$ , are divisible by none of 2, 5 and 7?

A.
41
B.
42
C.
40
D.
43

Answer

A

Explanation

Using Principle of Inclusion-Exclusion for $n = 120$ : $A$ : divisible by 2 $\implies |A| = 60$ $B$ : divisible by 5 $\implies |B| = 24$ $C$ : divisible by 7 $\implies |C| = 17$

$|A \cap B| = \lfloor 120/10 \rfloor = 12$ $|B \cap C| = \lfloor 120/35 \rfloor = 3$ $|A \cap C| = \lfloor 120/14 \rfloor = 8$ $|A \cap B \cap C| = \lfloor 120/70 \rfloor = 1$

Total divisible by at least one = $60 + 24 + 17 - (12 + 3 + 8) + 1 = 101 - 23 + 1 = 79$ . Number of integers divisible by none = $120 - 79 = 41$ .

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