Question

Find the least number which when divided by 15, leaves a remainder of 5, when divided by 25, leaves a remainder of 15 and when divided by 35 leaves a remainder of 25.
(a) 515
(b) 525
(c) 1040
(d) 1050

Answer 1

Hint: For solving this question we will use a basic concept of mathematics that if a number $n$ leaves remainder $r$ when divided by a number $q$ then, $n+q-r$ will be a multiple of $q$ . After that, we will find the L.C.M. of 15, 25 and 35 and solve for the correct value of the required number.

Complete step-by-step answer:
Given:
We have to find the least possible number, which is when divided by 15, leaves a remainder of 5, when divided by 25, leaves a remainder of 15 and when divided by 35 leaves a remainder of 25.
Now, let $n$ is the required number. We know that if a number $n$ leaves remainder $r$ when divided by a number $q$ then, $n+q-r$ will be a multiple of $q$ . And it is given that when $n$ is divided by 15 then it leaves a remainder of 5 which means $n+10$ will be a multiple of 15. And it is given that when $n$ is divided by 25, leaves a remainder of 15 which means $n+10$ will be a multiple of 25. And it is given that when $n$ is divided by 35, leaves a remainder of 25 which means $n+10$ will be a multiple of 35. Thus, we conclude that $n+10$ will be a multiple of 15, 25 and 35.
Now, for the least value of $n$ the value of $n+10$ will be the L.C.M. of the 15, 25, 35. So, we will find the L.C.M. of 15, 25 and 35.
Now, we will find the L.C.M. of 15, 25 and 35 by the fundamental theorem of arithmetic in which first we express any number in terms of multiplication of prime numbers. Then, we will find the value of the least common multiple (L.C.M).
We can write, $15=1\times 3\times 5$ , $25=1\times 5\times 5$ and $35=1\times 5\times 7$ . Then, L.C.M. of 15, 25 and 35 can be determined by taking the highest power of every prime number in the given numbers. Then,
$L.C.M=1\times 3\times 5\times 5\times 7=525$
Now, we can write that value of $n+10=525$ . Then,
$\begin{align}
  & n+10=525 \\
 & \Rightarrow n=515 \\
\end{align}$
Thus, the 515 is the required number.
Hence, (a) is the correct option.

Note: Here, the student should first understand what is asked in the question before solving the question. After that, we should figure out that if a number $n$ leaves remainder $r$ when divided by a number $q$ then, $n+q-r$ will be a multiple of $q$ and we should apply this concept correctly. Moreover, we should find L.C.M. correctly and calculate the required number correctly.