Question

Determine the smallest 3-digit number which is exactly divisible by 5, 8, 12.

Answer 1

Hint: The given question is related to the Least Common Multiple of three numbers. The Least Common Multiple of two or more numbers is the smallest number that is divisible by the given numbers. Find the LCM of 5, 8 and 12, then find the smallest 3-digit number divisible by the LCM, which will be the required answer.

Complete step-by-step answer:
Before proceeding with the problem, let’s understand the concept of LCM. LCM of two or more numbers is the smallest number that is divisible by all of the given numbers. For example: 6 is the LCM of 2 and 3. 6 is the smallest number that is divisible by both 2 and 3.

Now, coming to the solution, we are asked to find the smallest 3-digit number that is divisible by 5, 8 and 12. First, we will find the least common multiple of 5, 8 and 12. To find the LCM, we will use the method of prime factorization. We will express each number as the product of their prime factors. Then we will find the LCM using the prime factors.

5 is a prime number, so it will be a prime factor of itself.

We can write 8 as $8=2\times 2\times 2$ . We can write 12 as $12=2\times 2\times 3$ . So, the LCM of 5, 8 and 12 is $2\times 2\times 2\times 3\times 5=120$ . Now, we can see that the LCM is a 3-digit number. Hence, the smallest 3-digit number, which is exactly divisible by 5, 8 and 12 is 120.

Note: While calculating HCF and LCM, prime factorization is the easiest method. But it takes time. Hence, other methods should also be learnt, so that they can be used while solving problems in cases where time plays an important role.