multiples of 36 and 42

Answer 1

To find the multiples of two numbers, such as 36 and 42, you can look for their least common multiple (LCM). The multiples of both numbers are the numbers that can be expressed as \( n \times 36 \) and \( m \times 42 \), where \( n \) and \( m \) are integers.

Finding the LCM of 36 and 42:

1. Prime Factorization:

   • 36 = \( 2^2 \times 3^2 \)
   • 42 = \( 2^1 \times 3^1 \times 7^1 \)

2. Take the highest power of each prime:

   • From 36: \( 2^2 \), \( 3^2 \)
   • From 42: \( 2^1 \), \( 3^1 \), \( 7^1 \)

3. Combine:

   • \( 2^2 \) (highest from 36), \( 3^2 \) (highest from 36), \( 7^1 \) (from 42)

4. LCM Calculation:

   • LCM = \( 2^2 \times 3^2 \times 7^1 = 4 \times 9 \times 7 = 252 \)

Multiples of 36 and 42:

• The multiples of 36 are: 36, 72, 108, 144, 180, 216, 252, 288, 324, ...
• The multiples of 42 are: 42, 84, 126, 168, 210, 252, 294, 336, 378, ...

Common Multiples:

• The common multiples of 36 and 42 include:
  • 252, 504, 756, ...

So, the least common multiple of 36 and 42 is 252, and some of their common multiples are 252, 504, and so on.