Question

what is the smallest 3 - digit integer that is a multiple of 6 and divisible by 8? a 102 b 104 c 120 d 144

Explanation:

Step1: Find LCM of 6 and 8

Prime factorize 6: \(6 = 2\times3\)
Prime factorize 8: \(8 = 2^3\)
LCM is the product of highest powers: \(LCM(6,8)=2^3\times3 = 24\)

Step2: Find smallest 3 - digit multiple of 24

Divide 100 by 24: \(100\div24 = 4\) with remainder \(4\)
So the first multiple of 24 greater than 100 is \(24\times(4 + 1)=24\times5 = 120\)

Step3: Verify options

• Option A: \(102\div24 = 4.25\) (not divisible by 24)
• Option B: \(104\div24\approx4.33\) (not divisible by 24)
• Option C: \(120\div24 = 5\) (divisible by 24, so multiple of 6 and 8)
• Option D: 144 is a multiple but not the smallest 3 - digit one.

Answer:

C. 120