If the LCM of two Prime numbers is 437, Find one number given that the other is 23
The Correct Answer and Explanation is:
To find one of the numbers, we can use the relationship between the Least Common Multiple (LCM) and the numbers involved. The LCM of two numbers is given by the formula:LCM(a,b)=a×bGCD(a,b)\text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)}LCM(a,b)=GCD(a,b)a×b
Where:
- aaa and bbb are the two numbers.
- GCD is the Greatest Common Divisor of the two numbers.
Since 23 is a prime number, it only has two divisors: 1 and 23. Therefore, the GCD of 23 and any other prime number (say ppp) will always be 1, because prime numbers don’t share any divisors other than 1. This simplifies the LCM formula to:LCM(23,p)=23×p\text{LCM}(23, p) = 23 \times pLCM(23,p)=23×p
We are given that the LCM of the two numbers is 437. Therefore:23×p=43723 \times p = 43723×p=437
To solve for ppp, divide both sides of the equation by 23:p=43723p = \frac{437}{23}p=23437p=19p = 19p=19
Thus, the other number is 19.
Explanation:
- The LCM of two prime numbers is simply the product of the two numbers since their only common factor is 1.
- Given that one number is 23 and the LCM is 437, we divide the LCM by 23 to find the second number.
- Therefore, the other number is 19.
This solution assumes that both numbers are prime, as indicated in the question.
