Question

Find the greatest number that will divide 152, 245 and 616 leaving remainders 8, 5 and 4 respectively.
(A) 6
(B) 8
(C) 12
(D) 16

Answer 1

Hint: Subtract the given remainders from each of the respective numbers.
Find the greatest common divisor (GCD) of the numbers thus obtained. This will give the required answer.

Complete step-by-step answer:
We are given three numbers 152, 245, and 616.
We have to find the greatest number that will divide these numbers and also leave remainders 8, 5 and 4 respectively.
We are aware that we can express a number in terms of its remainder with the help of the following formula:
Dividend $ = $ Divisor $ \times $ Quotient $ + $ Remainder.
Therefore, remainder $ = $ Dividend $ - $ Divisor $ \times $ Quotient.
Let us apply this to each of the given numbers.
1) The remainder of 152 is given to be 8 if divided by a certain number.
Therefore, \[152 - 8 = 144 = \] Divisor $ \times $ Quotient.
2) The remainder of 245 is given to be 5 if divided by a certain number.
Therefore, \[245 - 5 = 240 = \]Divisor $ \times $ Quotient.
3) The remainder of 616 is given to be 4 if divided by a certain number.
Therefore, \[616 - 4 = 612 = \]Divisor $ \times $ Quotient.
Now, the divisor must be common to each of these numbers and it should also be the greatest possible divisor.
That is, we are looking for the greatest common divisor (GCD) of 144, 240, and 612.
To find the GCD, we need to consider the factors of each number and then look for the common ones among these.
The greatest of such common factors will be our required answer.
So, let’s write down the factors of 144, 240, and 612.
 $ \Rightarrow $ The factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144.
 $ \Rightarrow $ The factors of 240 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 80, 120, and 240.
 $ \Rightarrow $ The factors of 612 are 1, 2, 3, 4, 6, 12, 17, 18, 34, 36, 51, 102, 153, 206, 306, and 612.
Therefore, the common factors would be 1, 2, 3, 4, 6, and 12.
And the greatest among these factors is 12.
 $ \Rightarrow $ Hence the GCD of 144, 240, and 612 is 12.
Also,
 $
\Rightarrow 152 = 12 \times 12 + 8 \\
\Rightarrow 245 = 12 \times 20 + 5 \\
\Rightarrow 616 = 12 \times 51 + 4 \\
  $
Thus, 12 satisfies the given conditions and is the required answer.
Hence 12 is the greatest number that will divide 152, 245 and 616 leaving remainders 8, 5 and 4 respectively.

So, the correct answer is “Option B”.

Note:Some students tend to interpret the question wrongly and head for finding the GCD of the given numbers. However, we do not want the divisor to completely divide the numbers as we require the remainders. Therefore, this would be a completely wrong step.