To find the ones digit of \( x \) given the conditions of the problem, we can follow these steps:
### Step 1: Understand the conditions
We have two conditions:
1. When \( x \) is divided by 5, the remainder is 4.
2. When \( x \) is divided by 2, the remainder is 1.
### Step 2: Analyze the first condition
From the first condition, \( x \div 5 \) leaves a remainder of 4. This means:
\[ x = 5k + 4 \]
for some integer \( k \).
The possible values for \( x \) that satisfy this condition can be:
- If \( k = 0 \), then \( x = 4 \)
- If \( k = 1 \), then \( x = 9 \)
- If \( k = 2 \), then \( x = 14 \)
- If \( k = 3 \), then \( x = 19 \)
- If \( k = 4 \), then \( x = 24 \)
- And so on...
The ones digits of these values are: 4, 9, 4, 9, 4, ...
### Step 3: Analyze the second condition
From the second condition, \( x \div 2 \) leaves a remainder of 1. This means:
\[ x = 2m + 1 \]
for some integer \( m \).
The possible values for \( x \) that satisfy this condition can be:
- If \( m = 0 \), then \( x = 1 \)
- If \( m = 1 \), then \( x = 3 \)
- If \( m = 2 \), then \( x = 5 \)
- If \( m = 3 \), then \( x = 7 \)
- If \( m = 4 \), then \( x = 9 \)
- If \( m = 5 \), then \( x = 11 \)
- And so on...
The ones digits of these values are: 1, 3, 5, 7, 9, 1, ...
### Step 4: Find common values
Now we need to find the common ones digit from the two sets of values we derived:
- From the first condition, the ones digits are: 4, 9, 4, 9, ...
- From the second condition, the ones digits are: 1, 3, 5, 7, 9, ...
The common ones digit is **9**.
### Conclusion
Thus, the ones digit of \( x \) is **9**.
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