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

The correct option is A

$515$

Solution:

Step 1: Taking LCM of the three Quotients:

$$\begin{gathered} \mathrm{LCM}(15,25,35)=3\times 5\times 5\times 7 \\ \Rightarrow \mathrm{LCM}(15,25,35)=525 \end{gathered}$$

Step 2: Investigating the three cases:

According to the question statement,

1. When a number is divided by $15$, it leaves a remainder of $5$.
2. When a number is divided by $25$, leaves a remainder of $15$.
3. When a number is divided by $35$, leaves a remainder of $25$.

Step 3: Making an observation:

Now, $525$ divided by $35$ leaves a remainder $0$ but when $10$ is subtracted from $525$, it leaves a remainder $35-10=25$.

And, $525$ divided by $25$ leaves a remainder $0$ but when $10$ is subtracted from $525$, it leaves a remainder $25-10=15$.

And, $525$ divided by $15$ leaves a remainder $0$ but when $10$ is subtracted from $525$, it leaves a remainder $15-10=5$.

Hence, the least number required is $525-10=\mathbf{515}$

Final answer: Option (A) is correct.