If m is prime, n is a composite number and m + n= 240 also their LCM is 4199. Find m and n with explanation.
(a) 13, 227 (b) 17, 223 (c) 19, 221 (d) 23, 217
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Answers
Given:
Two numbers m and n such that m is a prime number and n is a composite number. The value of m + n is 240 and the LCM of m and n is 4199.
To Find:
The values of m and n are?
Solution:
1. The given numbers are m and n such that m is prime and n is composite.
2. A number is considered as a prime if it has only two factors that are 1 and the number itself.
For example, the number 13 has factors 1 and 13. Hence, 13 is a prime number.
3. A number is considered to be composite if it has more than two factors.
For example, the number 15 has factors 1, 3, 5, and 15. Hence, 15 is a composite number.
4. The sum of m and n is 240,
=> m + n = 240. ( Assume as equation 1 ).
5. The LCM of two numbers is defined as the least common multiple that is common between the two numbers.
- The LCM of a prime number and a composite number is the product of the two numbers.
6. The number m is prime and n is composite, Hence the LCM of m and n is m x n,
=> m x n = 4199.
7. The value of m and n can be calculated as follows,
=> (m+n)² = m² + n² + 2mn,
=> It can be also written as,
=> (m+n)² = m² + n² -2mn + 4mn,
=> (m+n)² = (m-n)² + 4mn,
=> (m+n)² - 4mn = (m-n)².
=> (240)² - 4(4199) = (m-n)²,
=> 57600 - 16796 = (m-n)²,
=> 40804 = (m-n)²,
=> √(40804) = m-n,
=> m-n = 202. ( Assume as equation 2 ).
8. Solve equations 1 and 2 for the values of m and n,
=> Add equations 1 and 2,
=> m + n + m - n = 202 + 240,
=> 2m = 442,
=> m = 221.
9. Substitute the value of m in equation 1,
=> 221 + n = 240,
=> n = 19.
Therefore, the values of m and n are 221 and 19 respectively. Option B is the correct answer.