Question

Find the sum of natural numbers \(a\) and \(b\). Statement (I): \(a>1,\; b>1\) Statement (II): \(\text{LCM}(a,b)=667\)

• A. Statement I alone is sufficient, but Statement II alone is not sufficient.
• B. Statement II alone is sufficient, but Statement I alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Even both statements together are not sufficient.

Hint:

Whenever the LCM is a product of distinct primes, first factorize it. The prime decomposition often uniquely determines the numbers.

Answer 1

The Correct Option is B

Concept: The Least Common Multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers. First factorize: \[ 667=23\times 29. \] Since both 23 and 29 are prime numbers, the divisors of 667 are: \[ 1,\;23,\;29,\;667. \] To have LCM equal to 667, the prime factors 23 and 29 must together appear in the numbers.

Step 1: Analyze Statement (I). Given: \[ a>1,\quad b>1. \] No numerical information is available. Infinitely many pairs satisfy this condition. Therefore Statement (I) alone is insufficient.

Step 2: Analyze Statement (II). Given: \[ \text{LCM}(a,b)=667. \] Since \[ 667=23\times 29, \] and both factors are prime, the only natural numbers greater than 1 whose LCM is 667 are: \[ a=23,\quad b=29 \] (or vice versa). Hence \[ a+b=23+29=52. \] A unique value is obtained. Therefore Statement (II) alone is sufficient.