Let N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then sum of the digits in N is

### Concept & Formula The greatest number $N$ dividing a set of numbers and leaving the same unknown remainder is determined by calculating the Highest Common Factor (HCF) of the absolute differences between the given numbers. ### Step-by-Step Solution * **Given:** The numbers are 1305, 4665, and 6905. * First, calculate the absolute pairwise differences: $$4665 - 1305 = 3360$$ $$6905 - 4665 = 2240$$ $$6905 - 1305 = 5600$$ * Now, find $N = \text{HCF of } (3360, 2240, 5600)$. * Let's inspect the numbers. They clearly end in zero, so 10 is a factor. Let's look at 336, 224, and 560. * Let's find the difference between the two closest values: $3360 - 2240 = 1120$. * The HCF must be a factor of 1120. Let's test if 1120 itself divides all three: $$3360 = 1120 \times 3$$ $$2240 = 1120 \times 2$$ $$5600 = 1120 \times 5$$ * Since 1120 cleanly divides all the differences, $N = 1120$. * The question asks for the sum of the digits in $N$. * Sum of digits = $1 + 1 + 2 + 0 = 4$. ### Exam Strategy & Shortcut You only need to calculate the differences between adjacent values: 3360 and 2240. The difference between them is $3360 - 2240 = 1120$. Simply test if 1120 is the HCF. It divides both, so it is the HCF. You can immediately sum the digits ($1+1+2+0=4$) without even calculating the third difference (5600) to save 10-15 seconds. ### Common Pitfall The most frequent error is misreading the final instruction. Students often correctly calculate $N = 1120$, look for 1120 in the options, panic when they don't see it, and guess blindly. Always underline exactly what you need to calculate ("sum of the digits in N"). ### Final Answer **Therefore, the correct answer is 4.**