Question

If two numbers, when divided by a divisor, gives remainder 35 and 30 respectively and when their sum is divided by the same divisor, the remainder is 20, then the divisor is:
(a) 40
(b) 45
(c) 50
(d) 55

Answer 1

Hint: Use Euclid division lemma to write the numbers in terms of its divisor, quotient, and remainder.

Complete step-by-step answer:
To start the question, let the numbers be x and y, respectively. Also, let the divisor be p.
According to Euclid’s Division lemma, if there are two integers s and t, there is always q and r for which the equation s=tq+r, such that r is positive and less than q.
So, the numbers can be written as;
x = pk+35……………..(i)
y = pl+30 ……………..(ii)
In the above equations, k and l are positive natural numbers. Now we will add both the equations as in the question a condition is given about the sum of the two numbers.
$x\text{ }+\text{ }y\text{ }=\text{ }pk\text{ }+\text{ }pl\text{ }+\text{ }65~~~~$
$\Rightarrow x\text{ }+\text{ }y\text{ }=\text{ }pn\text{ }+65~~~~$
But it is given in the question that the remainder should be 20, so the equation becomes:
$x\text{ }+\text{ }y\text{ }=\text{ }pn+\text{ }p+20~~~~$
Therefore, from the above results, we can say that p+20 is equal to 65. Writing this mathematically, we get
$p+20=65$
$\Rightarrow p=45$
Therefore, the divisor is 45, and the answer to the above question is option (b).

Note: Be careful about the calculation and the signs in the equation of Euclid’s division lemma. Also, see that the remainder is less than the divisor as this is a necessary condition for Euclid’s division lemma to be valid.