Find the value of odd natural number x if LCM (x,40)=1400.
To find the value of the odd natural number x, we first need to determine the prime factorization of 1400 and analyze its factors.
Step 1: Prime factorization of 1400
To find the prime factorization of 1400, we divide it by the smallest prime number, which is 2.
1400 ÷ 2 = 700
Further, we divide 700 by 2 again:
700 ÷ 2 = 350
Once more for 350:
350 ÷ 2 = 175
We can see that we can't divide 175 by 2 any further, so we move on to the next prime number, which is 3. However, 175 is not divisible by 3.
Next, we try dividing 175 by 5:
175 ÷ 5 = 35
Now, we try dividing 35 by 5:
35 ÷ 5 = 7
At this point, we can see that we cannot divide 7 by any prime number, so the prime factorization of 1400 is:
2 × 2 × 2 × 5 × 5 × 7
Step 2: Analyzing the factorization and determining the odd natural number
From the prime factorization of 1400, we can see that it contains two factors of 2 (2 × 2) and no other prime factors of 2. Thus, the odd natural number x must not contain any factors of 2.
To find the LCM (Least Common Multiple) of x and 40, we need to consider their prime factorizations, considering only the primes that are not common to both x and 40.
Prime factorization of 40:
40 ÷ 2 = 20
20 ÷ 2 = 10
10 ÷ 2 = 5
The prime factorization of 40 is: 2 × 2 × 2 × 5
Comparing the prime factorizations of 1400 and 40, we can see that the common factor is 2 × 2 × 2 × 5 = 40.
To calculate the LCM, we multiply the common factors by the remaining unique factors:
LCM (x, 40) = (2 × 2 × 7) × (2 × 2 × 2 × 5) = 28 × 40 = 1120
Since LCM (x, 40) is given as 1400, and x shouldn't have any factors of 2, the value of x can be calculated by dividing 1400 by 40:
x = 1400 ÷ 40 = 35
Therefore, the value of the odd natural number x is 35.