Question

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

Answer 1

Hint: We’ll first use the remainder theorem which is $N = pq + r$, where N is divided by p, the q and r come out to be the quotient and the remainder respectively, we’ll apply remainder theorem on all the three numbers given to us. From the remainder theorem, we’ll get the 3 multiples of the required number and from those multiples, after finding the HCF of those numbers we’ll approach to our answer.

Complete step by step solution: Given data: 122, 150 and 115 leaves remainder 5, 7, and 11 respectively when divided by the required number.
Let the integer be A
The remainder theorem states that if a number N is divided by p, the q and r comes out to be the quotient and the remainder respectively then the relation between them is
i.e. $N = pq + r$
According to the given data when 122 is divided by A leaves remainder 5
Let the quotient be ${q_1}$
$ \Rightarrow 122 = A{q_1} + 5$
On subtracting 5 from both sides, we get,
$ \Rightarrow 117 = A{q_1}$
And when 150 is divided by A leaves remainder 7
Let the quotient be ${q_2}$
$ \Rightarrow 150 = A{q_2} + 7$
On subtracting 7 from both sides we get,
$ \Rightarrow 143 = A{q_2}$
And when 115 is divided by A leaves the remainder 11
Let the quotient be ${q_3}$
$ \Rightarrow 115 = A{q_3} + 11$
On subtracting 11 from both the sides we get,
$ \Rightarrow 104 = A{q_3}$
Now we can say that the HCF of $A{q_1}$, $A{q_2}$ and $A{q_3}$will be the greatest value of A
Using the prime factorization method
$ \Rightarrow 104 = 2 \times 2 \times 2 \times 13$
$ \Rightarrow 143 = 11 \times 13$
$ \Rightarrow 117 = 3 \times 3 \times 13$
Therefore the HCF will be 13
Therefore the greatest integer value for A will be 13

i.e. (C)13 is the correct option.

Note: Before using the remainder theorem we should acknowledge the terms like divisors dividend quotient remainder.
1) The divisor is the number that divides.
2) The dividend is the number from which the number is being been divided.
3) The quotient is referred as to the number of times the divisor divides the dividend
4) The remainder is referred to as the residual number after the divisor divides the dividend.