Which greatest number will divide 3026 and 5053 leaving remainders 11 and 13 respectively?

### Concept & Formula When a question asks for the greatest number that divides given numbers $x$ and $y$ leaving specific remainders $a$ and $b$ respectively, the required number is the Highest Common Factor (HCF) of the numbers obtained by subtracting the remainders from the original numbers. Required Number = $\text{HCF of } (x - a) \text{ and } (y - b)$ ### Step-by-Step Solution **Given:** * Numbers: $3026$ and $5053$ * Respective Remainders: $11$ and $13$ **Calculation:** * Step 1: Subtract the respective remainders from the given numbers. * $3026 - 11 = 3015$ * $5053 - 13 = 5040$ * Step 2: Find the HCF of the resulting numbers ($3015$ and $5040$). * Step 3: Prime factorize or use divisibility rules. Let's look at the numbers. * $3015$ ends in $5$, so it's divisible by $5$. The sum of digits $(3+0+1+5=9)$ is divisible by $9$. Thus, it is divisible by $45$. * $3015 \div 45 = 67$. Since $67$ is a prime number, the prime factorization of $3015$ is $45 \times 67$. * Step 4: Check if the other number, $5040$, is divisible by these factors. * $5040$ is clearly divisible by $9$ (sum is $9$) and by $5$ (ends in $0$), so it is also divisible by $45$. * $5040 \div 45 = 112$. * Step 5: The factors of the two numbers are $(45 \times 67)$ and $(45 \times 112)$. Since $67$ and $112$ share no common factors, the HCF is $45$. ### Exam Strategy & Shortcut Leverage the options and basic divisibility rules. We need the HCF of $3015$ and $5040$. Since $3015$ ends in an odd digit ($5$), it cannot be divided by an even number. This immediately eliminates options (b) $30$ and (d) $60$. Now check the remaining options: $15$ and $45$. Apply the divisibility rule for $9$ (sum of digits). The sum of digits for both $3015$ and $5040$ is $9$, meaning both are divisible by $9$. Since $15$ is not a multiple of $9$ but $45$ is, the HCF must be $45$. ### Common Pitfall Students sometimes reverse the remainders (subtracting $13$ from $3026$ and $11$ from $5053$). Always ensure you map the remainders strictly in the "respective" order given in the problem statement. ### Final Answer **Therefore, the correct answer is 45.**