What is the numbers between 100 and 400 divisible by 2,3,5 and 7?Select one:a. 210b. 490c. 180d. 145e. 280

1. Break Down the Problem

To solve the problem, we need to find numbers that are:

• Between 100 and 400
• Divisible by 2, 3, 5, and 7

2. Relevant Concepts

A number that is divisible by multiple numbers must also be divisible by their least common multiple (LCM). Therefore, we first calculate the LCM of 2, 3, 5, and 7.

3. Calculate the LCM

• The prime factorization of the numbers:
  • $2 = 2^1$
  • $3 = 3^1$
  • $5 = 5^1$
  • $7 = 7^1$

The LCM is obtained by taking the highest power of each prime factor: $\text{LCM} = 2^1 \times 3^1 \times 5^1 \times 7^1 = 210$

4. Finding Multiples of the LCM

Now we need to find multiples of 210 that are between 100 and 400. We do this by dividing 100 and 400 by 210 and finding the integers:

• Smallest integer $n$ such that $210n > 100$:

  • $n = 1 \Rightarrow 210 \times 1 = 210$

• Largest integer $n$ such that $210n < 400$:

  • $n = 1 \Rightarrow 210 \times 2 = 420$ (too large)
  • $n = 1 \Rightarrow 210 \times 1 = 210$ (valid)

5. Verify and Summarize

The only number we found between 100 and 400 that is divisible by 2, 3, 5, and 7 is 210.

Final Answer

Therefore, the correct option is: a. 210