Find an odd natural number x such taht LCM(x,392)=10584
10584/392 = 27
Do 27 and 392 have any common factors ?
27 = 3*3*3
392 = 2*2*2*7*7
No, they don't, so 27 and 392 have the LCM of 10584,
your odd natural number is 27
10584/392 = 27
Do 27 and 392 have any common factors ?
27 = 3*3*3
392 = 2*2*2*7*7
No, they don't, so 27 and 392 have the LCM of 10584,
your odd natural number is 27
To find an odd natural number x such that the LCM (least common multiple) of x and 392 is 10584, we can follow these steps:
Step 1: Prime factorize both 10584 and 392.
Prime factorization of 10584:
10584 = 2^3 * 3^2 * 7^2
Prime factorization of 392:
392 = 2^3 * 7^2
Step 2: Now, we need to find the highest power of each prime factor that is present in both numbers.
Comparison of prime factors:
- The highest power of 2 present in 10584 is 2^3 (as it appears in the prime factorization).
- The highest power of 2 present in 392 is also 2^3.
- The highest power of 3 present in 10584 is 3^2.
- The highest power of 3 present in 392 is 3^0 (as it does not appear in the prime factorization).
- The highest power of 7 present in both 10584 and 392 is 7^2.
Step 3: Multiply these highest powers together to get the LCM.
LCM(x,392) = 2^3 * 3^2 * 7^2 = 10584.
So, the odd natural number x that satisfies the given condition is 10584.
To find an odd natural number x such that LCM(x, 392) = 10584, we need to understand a few key concepts and steps.
Step 1: Prime Factorization
First, let's find the prime factorization of 392. To do this, we can divide 392 by the smallest prime number, which is 2. We continue dividing by 2 until we can no longer divide evenly. The resulting quotient is 196.
392 ÷ 2 = 196
Now, we repeat the process with the quotient, dividing by the smallest prime number until we can no longer divide evenly.
196 ÷ 2 = 98
Again,
98 ÷ 2 = 49
We have reached a prime number, 49, so we cannot divide any further. Therefore, the prime factorization of 392 is 2^3 × 7^2.
Step 2: Finding the LCM
Now that we have the prime factorization of 392, we can find the LCM by using the highest exponent of each prime factor.
For the LCM(x, 392) to equal 10,584, we compare the prime factorization of 392 and 10,584.
Prime factorization of 392: 2^3 × 7^2
Prime factorization of 10,584: 2^3 × 3^3 × 7^1
Comparing the two, we select the highest exponent for each prime factor:
Highest exponent of 2: 3
Highest exponent of 3: 3
Highest exponent of 7: 2
Therefore, the LCM of x and 392 is 2^3 × 3^3 × 7^2, which simplifies to 2^3 × 3^3 × 7^2 = 2,744.
Step 3: Find an Odd Natural Number x
To find an odd natural number x such that LCM(x, 392) = 10,584, we need to find a number that has different prime factors and exponents than 392.
To do this, we can choose an odd prime number. Let's choose 5.
Prime factorization of 5: 5^1
Since the prime factorization of 392 does not have 5 as a factor, we can add it to the factorization of x. Therefore, a possible value for x could be:
x = 5^1 = 5
So, an odd natural number x that satisfies LCM(x, 392) = 10,584 is x = 5.