Two numbers, both greater than 29, have H.C.F. 29 and L.C.M. 4147. The sum of the numbers is

### Concept & Formula When given the H.C.F. of two numbers, we can define the numbers as $Hx$ and $Hy$, where $H$ is the H.C.F., and $x$ and $y$ are co-prime integers. $$ \text{L.C.M.} = H \times x \times y $$ ### Step-by-Step Solution * **Given:** H.C.F. ($H$) = 29, L.C.M. = 4147. Both numbers are greater than 29. * **Calculation / Deduction:** Let the numbers be $29x$ and $29y$. * Using the formula: $29 \times x \times y = 4147$. * Solve for $xy$: $xy = 4147 \div 29 = 143$. * We must find co-prime factors of 143. By testing small primes, we see $143 = 11 \times 13$. * So, $x = 11$ and $y = 13$. * The numbers are $29 \times 11 = 319$ and $29 \times 13 = 377$. * Both are strictly greater than 29, satisfying the given condition. * Find the sum: $319 + 377 = 696$. ### Exam Strategy & Shortcut Use unit digit estimation to verify calculations quickly. For the sum $29(11 + 13) = 29 \times 24$. Look at the unit digits: $9 \times 4 = 36$. The final answer must end in a 6. This eliminates option (b). Between 666 and 696, since $30 \times 24 = 720$, the answer $29 \times 24$ must be just slightly less than 720 ($720 - 24 = 696$). ### Common Pitfall Ignoring the condition "both greater than 29". If a student factored 143 as $1 \times 143$, the numbers would be $29 \times 1 = 29$ and $29 \times 143 = 4147$. While their H.C.F. is 29 and L.C.M. is 4147, one number is exactly 29, violating the "greater than 29" rule. Always test the co-prime pair that are closer in value. ### Final Answer Therefore, the correct answer is **696**.