Question

Calculate the number of positive integers less than $100$, which are divisible by $3,5$, and $7$.

• A. None
• B. One
• C. Two
• D. Three
• E. Four

Answer 1

The correct option is A None

We need to find the number($<100$) which are divisible by $3,5$ and $7$.

The numbers should be divisible by all three of them.

For this, we can take multiple of any of the $3,5$ or $7$ and check the divisibility of that multiples with other two remaining numbers.

Let's take the set of numbers which are divisible by $7$ and that are

$\{7,14,21,.....77,84,91,98\}$.

None of the above numbers are divisible by $5$.

Hence, none of the numbers($<100$) are divisible by all three numbers($3,5,7$).

(Alternate Method)
If you noticed that the three numbers are all different primes, then LCM (least common multiple) $=3\times 5\times 7=105$.

Since the LCM is greater than $100$, the answer is A.