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What is the least perfect square that is a multiple of 7,11 and 12?
Correct answer is '213444'. Can you explain this answer?
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What is the least perfect square that is a multiple of 7,11 and 12?Co...
Let us assume the least perfect square be X
⇒ 7 = 7 × 1
⇒ 11 = 11 × 1
⇒ 12 = 22 × 3
⇒ The LCM of (7,11,12) = 22 × 3 × 11 × 7
⇒ The least perfect square = 22 × 32 × 112 × 72 = 213444
Hence, the correct answer is 213444.
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What is the least perfect square that is a multiple of 7,11 and 12?Co...
Finding the Least Perfect Square
To find the least perfect square that is a multiple of 7, 11, and 12, we start by determining the least common multiple (LCM) of these numbers.
Step 1: Prime Factorization
- 7 is a prime number: 7 = 7^1
- 11 is a prime number: 11 = 11^1
- 12 can be factored into prime numbers: 12 = 2^2 × 3^1
Step 2: LCM Calculation
To find the LCM, we take the highest power of each prime factor:
- For 2: max power is 2 (from 12)
- For 3: max power is 1 (from 12)
- For 7: max power is 1 (from 7)
- For 11: max power is 1 (from 11)
Thus, the LCM is calculated as follows:
- LCM = 2^2 × 3^1 × 7^1 × 11^1 = 4 × 3 × 7 × 11
Step 3: Calculate the LCM
- 4 × 3 = 12
- 12 × 7 = 84
- 84 × 11 = 924
So, LCM(7, 11, 12) = 924
Step 4: Finding a Perfect Square
A perfect square must have even powers of all prime factors. The prime factorization of 924 is:
- 2^2 × 3^1 × 7^1 × 11^1
To make it a perfect square, we need:
- 3^1 to become 3^2 (add one more 3)
- 7^1 to become 7^2 (add one more 7)
- 11^1 to become 11^2 (add one more 11)
Thus, we multiply 924 by 3 × 7 × 11 = 231.
Step 5: Final Calculation
- Perfect square = 924 × 231 = 213444
Therefore, the least perfect square that is a multiple of 7, 11, and 12 is 213444.
To find the least perfect square that is a multiple of 7, 11, and 12, we start by determining the least common multiple (LCM) of these numbers.
Step 1: Prime Factorization
- 7 is a prime number: 7 = 7^1
- 11 is a prime number: 11 = 11^1
- 12 can be factored into prime numbers: 12 = 2^2 × 3^1
Step 2: LCM Calculation
To find the LCM, we take the highest power of each prime factor:
- For 2: max power is 2 (from 12)
- For 3: max power is 1 (from 12)
- For 7: max power is 1 (from 7)
- For 11: max power is 1 (from 11)
Thus, the LCM is calculated as follows:
- LCM = 2^2 × 3^1 × 7^1 × 11^1 = 4 × 3 × 7 × 11
Step 3: Calculate the LCM
- 4 × 3 = 12
- 12 × 7 = 84
- 84 × 11 = 924
So, LCM(7, 11, 12) = 924
Step 4: Finding a Perfect Square
A perfect square must have even powers of all prime factors. The prime factorization of 924 is:
- 2^2 × 3^1 × 7^1 × 11^1
To make it a perfect square, we need:
- 3^1 to become 3^2 (add one more 3)
- 7^1 to become 7^2 (add one more 7)
- 11^1 to become 11^2 (add one more 11)
Thus, we multiply 924 by 3 × 7 × 11 = 231.
Step 5: Final Calculation
- Perfect square = 924 × 231 = 213444
Therefore, the least perfect square that is a multiple of 7, 11, and 12 is 213444.
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What is the least perfect square that is a multiple of 7,11 and 12?Correct answer is '213444'. Can you explain this answer?
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