**Q52. Two numbers A and B are given**
What is $A + B$?

Given statements:
(i) LCM of $A$ and $B$ is 44 times their HCF
(ii) The sum of LCM of $A$ & $B$ and their HCF is 540
(iii) $\frac{A}{10} + \frac{B}{10}$ is an integer
(iv) $A + B 150$

Which of the given statements are redundant to find the answer of the question?

(a) statement (ii)
(b) statement (iii)
(c) statement (iv)
(d) statement (i)
(e) Answer cannot be determined even after using all the statements.

Question

**Q52. Two numbers A and B are given**  
What is $A + B$?

Given statements:  
(i) LCM of $A$ and $B$ is 44 times their HCF  
(ii) The sum of LCM of $A$ & $B$ and their HCF is 540  
(iii) $\frac{A}{10} + \frac{B}{10}$ is an integer  
(iv) $A + B > 150$

Which of the given statements are redundant to find the answer of the question?

(a) statement (ii)  
(b) statement (iii)  
(c) statement (iv)  
(d) statement (i)  
(e) Answer cannot be determined even after using all the statements.

Accepted Answer

**Solution:**

1. Let the HCF of $A$ and $B$ be $x$. Then from statement (i),
   - LCM of $A$ and $B$ = $44x$

2. From statement (ii), sum of LCM and HCF is 540:
   $$44x + x = 45x = 540 \implies x = \frac{540}{45} = 12$$

3. So,
   - HCF = 12
   - LCM = $44 	imes 12 = 528$

4. Since $A$ and $B$ have HCF $12$, we can write:
   $$A = 12a, \quad B = 12b$$
   where $a$ and $b$ are coprime integers.

5. Also, LCM is given by:
   $$	ext{LCM} = 12 	imes a 	imes b = 528 \implies a 	imes b = 44$$

6. The pairs of coprime integers $(a,b)$ such that $a 	imes b = 44$ are:
   - $(1, 44)$
   - $(4, 11)$

7. Therefore, possible sums:
   - $A + B = 12(a + b) = 12(1 + 44) = 12 	imes 45 = 540$
   - $A + B = 12(4 + 11) = 12 	imes 15 = 180$

8. From statement (iii), $\frac{A}{10} + \frac{B}{10}$ is an integer, so $A + B$ is divisible by 10.
   - Both 180 and 540 are divisible by 10, so this does not eliminate any possibility.

9. From statement (iv), $A + B > 150$.
   - Both 180 and 540 satisfy this condition.

10. Hence, even after using all statements, $A + B$ can be either 180 or 540.

**Answer:** (e) Answer cannot be determined even after using all the statements.

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