Find two natural numbers in ratio 5:7 such that their l.c.m.is 120
huh. the numbers are 5x and 7x, but 7 does not divide 120. ???
30, 42
To find two natural numbers in the ratio 5:7 such that their least common multiple (lcm) is 120, we can follow these steps:
Step 1: Write the ratio in fraction form.
The given ratio 5:7 can be written as \( \frac{5}{7} \).
Step 2: Assume the two natural numbers.
Let's assume the two natural numbers in the ratio are 5x and 7x, where x is a common factor.
Step 3: Find the lcm of the assumed numbers.
To find the lcm of 5x and 7x, we need to find the least common multiple of 5 and 7.
The prime factors of 5 are 5, and the prime factors of 7 are 7.
Since there are no common prime factors, the lcm of 5 and 7 is simply their product: 5 * 7 = 35.
Step 4: Solve for x.
Given that the lcm is 120, we need to multiply both 5x and 7x by a factor to obtain 120 as the lcm. That factor will be \( \frac{120}{35} \) since lcm(\(5 \times \frac{120}{35}\)) = 120.
Simplifying \( \frac{120}{35} \), we get \( \frac{24}{7} \).
Step 5: Calculate the numbers.
Multiplying the assumed numbers by the value of x (\( \frac{24}{7} \)), we get the two numbers.
The first number = \( 5x = 5 \times \frac{24}{7} = \frac{120}{7} \).
The second number = \( 7x = 7 \times \frac{24}{7} = 24 \).
Therefore, the two natural numbers in the ratio 5:7 such that their lcm is 120 are \( \frac{120}{7} \) and 24.