The product of two numbers is $4107$. If the H.C.F. of these numbers is $37$, then the greater number is

### Concept & Formula When the Highest Common Factor (H.C.F.) of two numbers is given, the numbers can be algebraically represented as $H \times a$ and $H \times b$, where $H$ is the H.C.F., and $a$ and $b$ are strictly **co-prime** integers (meaning they share no common factors other than $1$). The product of the two numbers is given by the formula: $$(H \times a) \times (H \times b) = \text{Product}$$ ### Step-by-Step Solution * **Given:** The H.C.F. ($H$) is $37$. The product of the two numbers is $4107$. * **Step 1:** Represent the numbers using the H.C.F. Let the two numbers be $37a$ and $37b$, where $a$ and $b$ are co-prime. * **Step 2:** Set up the equation using the given product. $$37a \times 37b = 4107$$ $$1369 \times ab = 4107$$ * **Step 3:** Solve for the product of the co-prime factors, $ab$. $$ab = \frac{4107}{1369}$$ $$ab = 3$$ * **Step 4:** Determine the possible co-prime pairs for $(a, b)$ that multiply to $3$. Since $3$ is a prime number, the only possible pair is $(1, 3)$. So, let $a = 1$ and $b = 3$. * **Step 5:** Calculate the actual numbers. First number: $37 \times 1 = 37$ Second number: $37 \times 3 = 111$ * **Step 6:** Identify the greater number. Between $37$ and $111$, the greater number is $111$. ### Exam Strategy & Shortcut **Divisibility & Logical Deduction:** You know the numbers must be multiples of $37$. Looking at the options, you are searching for a number that, when multiplied by another multiple of $37$, yields $4107$. Since the last digit of the product is $7$, and the H.C.F. ends in $7$, you can quickly test the unit digits. If the greater number is $111$ (ends in $1$), the smaller number must end in $7$ to make the product end in $7$ ($1 \times 7 = 7$). Since $37$ ends in $7$, $37 \times 111$ is a perfect candidate. $37 \times 111 = 4107$. The math checks out instantly. ### Common Pitfall A frequent mistake is forgetting to divide the product by the square of the H.C.F. ($H^2$). Students often divide $4107$ by just $37$ once, obtaining $111$, and mistakenly assume this is the product $ab$. While $111$ happens to be the final answer here by coincidence, this flawed method will yield completely incorrect results on almost any other exam question of this type. ### Final Answer **Therefore, the correct answer is 111.**