Question

The greatest number which can divide 309, 814 and 2219 leaving the remainders 3, 4 and 5 respectively is
(a) 16
(b) 17
(c) 18
(d) 19

Answer 1

Hint: In this problem, we are provided with some numbers leaving remainders corresponding to each number. By subtracting the remainder from the number, the number becomes exactly divisible. Now we have to find the greatest number which can divide all this number. So, the HCF of the given number is required.

Complete step-by-step answer:
According to the problem statement, we are given 309, 814 and 2219 which when divided by a number leaves the remainder 3, 4 and 5 respectively.
To obtain the exactly divisible number we subtract the reminders from the corresponding number.
$\begin{align}
  & \Rightarrow 309-3=306 \\
 & \Rightarrow 814-4=810 \\
 & \Rightarrow 2219-5=2214 \\
\end{align}$
The new set of numbers are 306, 810 and 2214.
As per the definition of HCF, the largest number that divides two or more numbers is the highest common factor (HCF) of a given set of numbers. So, the greatest number that can divide is the HCF of three numbers. To find the HCF of the given numbers, we express each number as a product of prime numbers. The highest prime factor is HCF.
By using the prime factorisation methodology, we get
$\begin{align}
  & 306=2\times 3\times 3\times 17 \\
 & \Rightarrow 306=2\times {{3}^{2}}\times 17 \\
 & 810=2\times 3\times 3\times 3\times 3\times 5 \\
 & \Rightarrow 810=2\times {{3}^{4}}\times 5 \\
 & 2214=2\times 3\times 3\times 3\times 41 \\
 & \Rightarrow 2214=2\times {{3}^{3}}\times 41 \\
\end{align}$
The common factors in the product are one 2 and two 3’s.
Therefore, the HCF of the given numbers is $2\times 3\times 3=18$.
Hence option (c) is correct.

So, the correct answer is “Option (c)”.

Note: This problem can be alternatively solved by using hit and trial method. After obtaining the exactly divisible numbers, we individually check all the options to obtain the correct answer. Like 810 is not divisible by 16, so 16 is rejected. 810 is also not divisible by 17, so 17 is rejected. Now considering 18, all the numbers are divisible. So, this is the correct answer.