When an integer K is divided by 3, the remainder is 1, and when K+ 1 is divided by 5, the remainder is 0. Of the following, a possible value of K is

Options

A. 62
B. 63
C. 64
D. 65

Answer

To solve the problem step by step, we need to analyze the conditions given for the integer \( K \).

### Step 1: Understand the first condition
The first condition states that when \( K \) is divided by 3, the remainder is 1. This can be expressed mathematically as:
\[ K \equiv 1 \ (\text{mod} \ 3) \]
This means that \( K \) can be written in the form:
\[ K = 3n + 1 \]
for some integer \( n \).

### Step 2: Understand the second condition
The second condition states that when \( K + 1 \) is divided by 5, the remainder is 0. This can be expressed as:
\[ K + 1 \equiv 0 \ (\text{mod} \ 5) \]
This means that:
\[ K + 1 = 5m \]
for some integer \( m \), or equivalently:
\[ K \equiv -1 \ (\text{mod} \ 5) \]
Since \(-1\) is equivalent to \(4\) in modulo 5, we can rewrite this as:
\[ K \equiv 4 \ (\text{mod} \ 5) \]

### Step 3: Set up the simultaneous congruences
Now we have two conditions:
1. \( K \equiv 1 \ (\text{mod} \ 3) \)
2. \( K \equiv 4 \ (\text{mod} \ 5) \)

### Step 4: Solve the simultaneous congruences
We can solve these congruences using the method of substitution or trial and error.

From the first condition, we can list some possible values of \( K \):
- If \( n = 0 \), then \( K = 1 \)
- If \( n = 1 \), then \( K = 4 \)
- If \( n = 2 \), then \( K = 7 \)
- If \( n = 3 \), then \( K = 10 \)
- If \( n = 4 \), then \( K = 13 \)
- If \( n = 5 \), then \( K = 16 \)
- If \( n = 6 \), then \( K = 19 \)
- If \( n = 7 \), then \( K = 22 \)
- If \( n = 8 \), then \( K = 25 \)
- If \( n = 9 \), then \( K = 28 \)
- If \( n = 10 \), then \( K = 31 \)
- If \( n = 11 \), then \( K = 34 \)
- If \( n = 12 \), then \( K = 37 \)
- If \( n = 13 \), then \( K = 40 \)
- If \( n = 14 \), then \( K = 43 \)
- If \( n = 15 \), then \( K = 46 \)
- If \( n = 16 \), then \( K = 49 \)
- If \( n = 17 \), then \( K = 52 \)
- If \( n = 18 \), then \( K = 55 \)
- If \( n = 19 \), then \( K = 58 \)
- If \( n = 20 \), then \( K = 61 \)
- If \( n = 21 \), then \( K = 64 \)

Now we can check which of these values satisfies the second condition \( K \equiv 4 \ (\text{mod} \ 5) \).

### Step 5: Check the values
Starting from \( K = 64 \):
- \( K + 1 = 65 \)
- \( 65 \div 5 = 13 \) with a remainder of 0.

Thus, \( K = 64 \) satisfies both conditions.

### Conclusion
A possible value of \( K \) is \( 64 \).

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