In a school, all the students can stand in a row, so that each row has 5, 9 or 10 students. Find the least number of students in the school.

Options

A. 90
B. 95
C. 85
D. 100

Answer

To find the least number of students in the school who can stand in rows of 5, 9, or 10, we need to calculate the least common multiple (LCM) of these three numbers. Here’s a step-by-step solution:

### Step 1: Identify the numbers
We have three numbers: 5, 9, and 10.

### Step 2: Prime factorization
We will find the prime factorization of each number:
- 5 is a prime number, so its factorization is \(5^1\).
- 9 can be factored as \(3^2\) (since \(9 = 3 \times 3\)).
- 10 can be factored as \(2^1 \times 5^1\).

### Step 3: Determine the LCM
To find the LCM, we take the highest power of each prime number that appears in the factorizations:
- The prime number 2 appears in \(10\) as \(2^1\).
- The prime number 3 appears in \(9\) as \(3^2\).
- The prime number 5 appears in both \(5\) and \(10\) as \(5^1\).

Thus, the LCM is calculated as:
\[ LCM = 2^1 \times 3^2 \times 5^1 \]

### Step 4: Calculate the LCM
Now we will compute the LCM:
\[ LCM = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 \]
Calculating this step-by-step:
1. \(2 \times 9 = 18\)
2. \(18 \times 5 = 90\)

### Step 5: Conclusion
The least number of students in the school is 90.

### Final Answer
The least number of students in the school is **90**.

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