write all the numbers less than 100 which are common multiples of 2, 3 and 5

write all the numbers less than 100 which are common multiples of 2, 3 and 5

The correct answer and explanation is:

The correct answer is: 30, 60, 90

Explanation:

To find the common multiples of 2, 3, and 5, we first need to calculate the least common multiple (LCM) of these three numbers. The LCM is the smallest number that is a multiple of all three numbers.

1. Prime Factorization:
   • The prime factorization of 2 is 22.
   • The prime factorization of 3 is 33.
   • The prime factorization of 5 is 55.
2. LCM Calculation:
   The LCM is found by taking the highest power of each prime factor that appears in any of the numbers:
   • The highest power of 2 is 212^1.
   • The highest power of 3 is 313^1.
   • The highest power of 5 is 515^1.
   Therefore, the LCM of 2, 3, and 5 is: LCM(2,3,5)=2×3×5=30LCM(2, 3, 5) = 2 \times 3 \times 5 = 30
3. Common Multiples:
   Once we have the LCM, we can find all the multiples of 30 that are less than 100. These multiples are obtained by multiplying 30 by successive integers:
   • 30×1=3030 \times 1 = 30
   • 30×2=6030 \times 2 = 60
   • 30×3=9030 \times 3 = 90
   The next multiple of 30 is 30×4=12030 \times 4 = 120, which is greater than 100, so we stop here.

Thus, the common multiples of 2, 3, and 5 that are less than 100 are 30, 60, and 90.

Why LCM is Important:

The LCM helps in identifying numbers that can be evenly divided by multiple numbers. In this case, 30 is the smallest number that can be divided evenly by 2, 3, and 5, and the subsequent multiples (60 and 90) follow the same pattern.