Question

Find the largest positive integer that will divide 122,150,115 leaving remainder 5,7,11 respectively
A) 3
B) 2
C) 13
D) 11

Answer 1

Hint: We know that in long division method dividend = divisor x quotient + remainder. So we will find divisor x quotient by subtracting the given remainder from the respective numbers and then find the multiples of divisor x quotient and compare them. Product of all common multiples will be the largest number and it will be our answer i.e. their HCF of all three of them.

Complete step-by-step answer:
We will find the value of divisor x quotient for each number.
So,
     \[
  122 = a \times {q_1} + 5 \Rightarrow a \times {q_1} = 117 \\
  150 = a \times {q_2} + 7 \Rightarrow a \times {q_2} = 143 \\
  115 = a \times {q_3} + 11 \Rightarrow a \times {q_3} = 104 \\
\]
, where a is the required term and q1, q2, q3 are the values of quotients of respective divisions.
Clearly, a will be Highest Common Factor (HCF) of 104,117,143
     \[\begin{array}{*{20}{c}}
  {104}& = &2& \times &2& \times &2& \times &{\boxed{13}} \\
  {117}& = &3& \times &3& \times &{\boxed{13}}&{}&{} \\
  {143}& = &{11}& \times &{\boxed{13}}&{}&{}&{}&{}
\end{array}\]
Since in a factor of 104,117,143 only 13 is common so the largest common factor is 13 or HCF = 13 and since it is the largest number which is common to all divisor x quotient pairs so our answer is 13.

Option C 13 is the correct answer.

Note: This question can be done by option checking. You can divide the number by each option and check whether the remainder is equal to that required or not. For example if I suppose my answer is 3 then dividing by 122 remainder will be 0,which is wrong so option is rejected and likewise other options can be checked.