The sum of two numbers is $462$ and their highest common factor is $22$. What is the minimum number of pairs that satisfy these conditions?

### Concept & Logic If the Highest Common Factor (H.C.F.) of two numbers is $H$, the numbers can always be written as $Ha$ and $Hb$, where $a$ and $b$ are **co-prime** integers. Co-prime means they share no common factors other than $1$. If they shared another factor, it would increase the overall H.C.F., violating the premise of the question. ### Step-by-Step Solution **Step 1: Set up the algebraic equation.** * Given H.C.F. = $22$. * Let the required numbers be $22a$ and $22b$. * We are given that their sum is $462$: $$ 22a + 22b = 462 $$ **Step 2: Simplify the equation.** * Factor out $22$ and divide the entire equation: $$ 22(a + b) = 462 $$ $$ a + b = \frac{462}{22} $$ $$ a + b = 21 $$ **Step 3: Find all co-prime pairs summing to 21.** * We must systematically list pairs of numbers $(a, b)$ that add up to $21$, and filter out any pairs that are not co-prime. * $(1, 20)$ $\rightarrow$ Co-prime (Valid) * $(2, 19)$ $\rightarrow$ Co-prime (Valid) * $(3, 18)$ $\rightarrow$ Not co-prime (Both divisible by $3$) * $(4, 17)$ $\rightarrow$ Co-prime (Valid) * $(5, 16)$ $\rightarrow$ Co-prime (Valid) * $(6, 15)$ $\rightarrow$ Not co-prime (Both divisible by $3$) * $(7, 14)$ $\rightarrow$ Not co-prime (Both divisible by $7$) * $(8, 13)$ $\rightarrow$ Co-prime (Valid) * $(9, 12)$ $\rightarrow$ Not co-prime (Both divisible by $3$) * $(10, 11)$ $\rightarrow$ Co-prime (Valid) **Step 4: Count the valid pairs.** * The valid co-prime pairs are: $(1, 20), (2, 19), (4, 17), (5, 16), (8, 13)$, and $(10, 11)$. * Counting them, there are exactly $6$ such pairs. ### Exam Strategy & Shortcut To quickly find co-prime pairs for a sum, list out the pairs systematically starting from $1$. Whenever you encounter a pair where both numbers share a factor (like both are even, or both are multiples of $3$), cross it out immediately. Being fast at recognizing common divisibility (especially by $2$ and $3$) is key to solving this rapidly. ### Common Pitfall The most dangerous mistake is forgetting the **co-prime** rule. Students often count all possible pairs that sum to $21$ (which would be $10$ pairs) and get the wrong answer. Remember, if $a$ and $b$ aren't co-prime, their common factor would scale up the H.C.F. beyond the stated $22$. ### Final Answer **Therefore, the correct answer is $6$.**