Question

The greatest number which divides $261$ ,$933$ and $1381$ leaving remainder $5$ in each case is
$
  A.\,\,\,128 \\
  B.\,\,\,64 \\
  C.\,\,\,32 \\
  D.\,\,\,16 \\
 $

Answer 1

Hint: We have to find the greatest common divisor (GCD) that is Highest common factor (HCF) of the given three numbers which on division leaves $5$ as remainder.
So, first we have to get the three numbers by subtracting the remainder, that is from each and then finding the HCF of the resulting numbers.

Complete step-by-step answer:
According to question, we are given three numbers as and $1381$
These are when divided by their greatest common divisor leaves remainder as $5$ ,
And we have to find that Greatest Common Divisor.
So, firstly we will subtract the remainder that is 5 from each number and get the required numbers as
$261 - 5\,\, = \,\,256$ (Subtracting remainder from each number)
$933\, - \,5\, = \,928$
$1381\, - \,5\, = \,1376$
Thus we get the numbers as \[\]
Now, we will find the HCF of each of these numbers.
That is $HCF\left( {256,928,1376} \right)$
Thus we get the factors of each number as-
$256\,\,\, = \,\,2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
$928\, = \,2 \times 2 \times 2 \times 2 \times 2 \times 29$
$1376\, = \,2 \times 2 \times 2 \times 2 \times 2 \times 43$
Thus common factors in all three numbers are
Thus the greatest number which divides leaving the remainder $5$ is $32$.

Additional Information: HCF and GCD are the same. And HCF can also be found by other methods. One method is to divide one number with another by taking the greater number as dividend and smaller number as divisor. Then obtained quotient is again divided by the third number and resultant quotient is the required HCF of the three numbers.

Note: Mistake could be made while not subtracting the remainder from the given numbers and finding the required numbers for HCF. Students should take care to first subtract the remainder and then find the HCF of the resulting number; otherwise we will get the wrong answer.