LCM of 23, 32 and 22, 33?

To find the least common multiple (LCM) of the given numbers, we first need to consider the prime factorization of each number:

• 23 is a prime number, so its factorization is simply 23.
• 32 can be expressed as 2⁵ (since 32 = 2 × 2 × 2 × 2 × 2).
• 22 can be factored into primes as 2 × 11.
• 33 factors into 3 × 11.

Now, let’s write down the prime factorization:

• 23 = 23¹
• 32 = 2⁵
• 22 = 2¹ × 11¹
• 33 = 3¹ × 11¹

To find the LCM, we take the highest power of each prime that appears in any of the numbers:

• Highest power of 2 is 2⁵ (from 32).
• Highest power of 3 is 3¹ (from 33).
• Highest power of 11 is 11¹ (from either 22 or 33).
• Highest power of 23 is 23¹ (from 23).

Now, we combine these to find the LCM:

LCM = 2⁵ × 3¹ × 11¹ × 23¹

Calculating this gives:

LCM = 32 × 3 × 11 × 23

Now, we can simplify this multiplication to find the final answer. Since we are mainly interested in the factors that were asked about in the options, we can see:

• We indeed have 23 and 32 as factors involved in the LCM computation.
• Therefore, the LCM of 23, 32, 22, and 33 could be represented concisely as the product of those relevant factors.

From the options given, the correct one that represents factors in the LCM would be:

• d) 22, 32 (since both 22 and 32 contribute in part to the full LCM).

Hence, the answer is option d) 22, 32.