On dividing a number by 13, we get 1 as remainder. If the quotient is divided by 5. we get 3 as a remainder. If this number is divided by 65, what will be the remainder ?

Options

A. 28
B. 18
C. 16
D. 40

Answer

To solve the problem step by step, we will define the number and use the information given in the question to find the remainder when this number is divided by 65.

### Step 1: Define the number
Let the number be \( D \).

### Step 2: Use the first condition
According to the problem, when \( D \) is divided by 13, the remainder is 1. This can be expressed as:
\[ D = 13q + 1 \]
where \( q \) is the quotient when \( D \) is divided by 13.

### Step 3: Use the second condition
Next, we know that when the quotient \( q \) is divided by 5, the remainder is 3. This can be expressed as:
\[ q = 5q' + 3 \]
where \( q' \) is the new quotient when \( q \) is divided by 5.

### Step 4: Substitute \( q \) into the equation for \( D \)
Now, we can substitute the expression for \( q \) from Step 3 into the equation for \( D \):
\[ D = 13(5q' + 3) + 1 \]
Expanding this gives:
\[ D = 65q' + 39 + 1 \]
\[ D = 65q' + 40 \]

### Step 5: Find the remainder when \( D \) is divided by 65
Now, we need to find the remainder when \( D \) is divided by 65. From the equation \( D = 65q' + 40 \), we can see that when \( D \) is divided by 65, the term \( 65q' \) will be divisible by 65, and we are left with the remainder:
\[ \text{Remainder} = 40 \]

### Conclusion
Thus, when the number \( D \) is divided by 65, the remainder is **40**.

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