How many positive integers less than 100 are divisible by both 4 and 6, but are NOT divisi… — GRE General Test

How many positive integers less than $100$ are divisible by both $4$ and $6$ , but are NOT divisible by $8$ ?

Cevap: 4

Cevap

The correct numerical answer is 4.

To be divisible by both 4 and 6, an integer must be a multiple of

\text{LCM}(4, 6) = 12

. The positive integers less than 100 that are multiples of 12 are 12, 24, 36, 48, 60, 72, 84, and 96 (8 integers). Among these, those divisible by 8 are multiples of

\text{LCM}(12, 8) = 24

, which are 24, 48, 72, and 96 (4 integers). Subtracting the excluded integers yields

8 - 4 = 4

Adım Adım Çözüm

Find the least common multiple of 4 and 6.

LCM(4, 6) = 12

An integer divisible by both 4 and 6 must be a multiple of their least common multiple.

Count positive integers less than 100 that are multiples of 12.

The multiples are 12, 24, 36, 48, 60, 72, 84, and 96, giving 8 integers.

The largest multiple of 12 strictly less than 100 is 96 (12 × 8).

Identify multiples of 12 that are also divisible by 8.

Since LCM(12, 8) = 24, these are the multiples of 24: 24, 48, 72, and 96, giving 4 integers.

Any integer divisible by both 12 and 8 must be a multiple of 24.

Subtract the excluded integers from the total count.

8 - 4 = 4

We exclude the multiples of 8 from the set of multiples of 12.

Anahtar Kavram

Divisibility, Least Common Multiple (LCM), and Set Exclusion

Soru

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Adım Adım Çözüm

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