To solve the problem, we will follow these steps:
### Step 1: Identify the Sample Space
The sample space consists of the first 25 natural numbers, which are:
\[ S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25\} \]
### Step 2: Identify Composite Numbers
Composite numbers are numbers that have more than two factors. The composite numbers from 1 to 25 are:
\[ \{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25\} \]
This gives us a total of 15 composite numbers.
### Step 3: Identify Non-Composite Numbers
Non-composite numbers include prime numbers and the number 1. The non-composite numbers from 1 to 25 are:
\[ \{1, 2, 3, 5, 7, 11, 13, 17, 19, 23\} \]
This gives us a total of 10 non-composite numbers.
### Step 4: Determine Divisibility Conditions
- If a composite number is selected, it is divided by 5.
- If a non-composite number is selected, it is divided by 2.
### Step 5: Find Composite Numbers Divisible by 5
The composite numbers from our list that are divisible by 5 are:
\[ \{10, 15, 20, 25\} \]
This gives us a total of 4 favorable outcomes from composite numbers.
### Step 6: Find Non-Composite Numbers Divisible by 2
The only non-composite number from our list that is divisible by 2 is:
\[ \{2\} \]
This gives us a total of 1 favorable outcome from non-composite numbers.
### Step 7: Calculate Total Favorable Outcomes
The total number of favorable outcomes (no remainder in division) is:
\[ 4 \text{ (from composites)} + 1 \text{ (from non-composites)} = 5 \]
### Step 8: Calculate Total Outcomes
The total number of outcomes in the sample space is:
\[ 25 \]
### Step 9: Calculate Probability
The probability of selecting a number such that there is no remainder in the division is given by:
\[
P(\text{no remainder}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{25} = \frac{1}{5}
\]
### Final Answer
Thus, the probability that there will be no remainder in the division is:
\[
\frac{1}{5} \text{ or } 0.2
\]
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