Three numbers which are co-prime to each other are such that the product of the first two is $551$ and that of the last two is $1073$. The sum of the three numbers is

### Concept & Logic Let the three numbers be $A$, $B$, and $C$. If they are all pairwise co-prime, it means no two numbers share a common factor greater than $1$. We are given two overlapping products: Product 1: $A \times B$ Product 2: $B \times C$ The middle number, $B$, is common to both products. Therefore, $B$ must be exactly equal to the Highest Common Factor (H.C.F.) of the two given product values. ### Step-by-Step Solution * **Given:** Let the three numbers be $x$, $y$, and $z$. $x \times y = 551$ $y \times z = 1073$ * **Step 1:** Identify the common variable. The number $y$ is a factor of both $551$ and $1073$. Since the numbers are co-prime to each other, $y$ is the *only* common factor between these two products. Thus, $y = \text{H.C.F.}(551, 1073)$. * **Step 2:** Calculate the H.C.F. of $551$ and $1073$. Using the long division method or difference method: Difference $= 1073 - 551 = 522$. The H.C.F. must be a factor of $522$. Let's test divisibility. $551$ is not divisible by $2$ or $3$. Let's use Euclidean division: $$1073 = 551 \times 1 + 522$$ $$551 = 522 \times 1 + 29$$ $$522 = 29 \times 18 + 0$$ The remainder is $0$, so the H.C.F. is $29$. Therefore, the middle number $y = 29$. * **Step 3:** Find the first number $x$. $$x \times 29 = 551$$ $$x = \frac{551}{29} = 19$$ * **Step 4:** Find the third number $z$. $$29 \times z = 1073$$ $$z = \frac{1073}{29} = 37$$ * **Step 5:** Calculate the sum of the three numbers. $$\text{Sum} = x + y + z = 19 + 29 + 37 = 85$$ ### Exam Strategy & Shortcut **Difference Method for Fast H.C.F.:** Finding the H.C.F. of large odd numbers can be tedious. Instead, find their difference: $1073 - 551 = 522$. The H.C.F. must divide this difference. $522$ is even, so divide it by $2 \rightarrow 261$. Divide by $3 \rightarrow 87$. Divide by $3$ again $\rightarrow 29$. $29$ is prime. Try dividing $551$ by $29$; it works perfectly ($19$). You now have $x=19$, $y=29$. Then find $z = 1073 / 29 = 37$. Add them up for $85$. ### Common Pitfall Students often waste immense amounts of time trying to brute-force prime factorize $551$ and $1073$ independently without realizing they share a massive common factor. Always look for the conceptual overlap (the middle number) and use the H.C.F. to unlock both factorizations simultaneously. ### Final Answer **Therefore, the correct answer is 85.**