Question

The least number which on division by 35 leaves a remainder 25 and on division by 45 leaves the remainder 35 and on division by 55 leaves the remainder 45 is:
(a). 2515
(b). 3455
(c). 2875
(d). 2785

Answer 1

Hint: Subtracting all the divisors with their corresponding remainders given in the question we get the difference as 10 so the required number is obtained by taking the L.C.M of 35, 45, 55 and then do the subtraction between the result of this L.C.M and 10.

Complete step-by-step answer:

In the question above, we have given three divisors and the remainder corresponding to those divisors. In the following we have shown the divisors and their corresponding remainders. The first number is divisor and the second number is its corresponding remainder.
 35, 25
45, 35
55, 45
We are going to find the difference of these divisors and remainders corresponding to the divisors.
$\begin{align}
  & 35-25=10 \\
 & 45-35=10 \\
 & 55-45=10 \\
\end{align}$
As you can see from the above that we are getting the same difference so to find the least number we have to take the L.C.M of 35, 45, 55 and then subtract 10 from the result of this L.C.M.
To find the L.C.M (35, 45, 55) we are going to write the factors of all the three numbers.
$\begin{align}
  & 35=7\times 5 \\
 & 45=3\times 3\times 5 \\
 & 55=11\times 5 \\
\end{align}$
Now, the L.C.M of the three numbers will be the multiplication of the least number which is common among the three numbers.
$\begin{align}
  & L.C.M\left( 35,45,55 \right)=5\times 9\times 7\times 11 \\
 & \Rightarrow L.C.M\left( 35,45,55 \right)=3465 \\
\end{align}$
Subtracting 10 from the result of the L.C.M of the three numbers we get,
$3465-10=3455$
From the above solution, the least number which on division by 35 leaves a remainder 25 and on division by 45 leaves the remainder 35 and on division by 55 leaves the remainder 45 is 3455.
Hence, the correct option is (b).

Note: You might be thinking why we have taken the L.C.M of the three divisors and then subtract with 10.
First of all we have to find the least number which on division with the 3 divisors gives the corresponding remainders so the least number is implying that we should take L.C.M of the three divisors and then we have subtracted from 10 to compensate the three different remainders.