Find the largest number that divides 92 and 75 and leave the remainders 2 and 5, respectively.

Options

A. 10
B. 15
C. 25
D. 30

Answer

To solve the problem of finding the largest number that divides 92 and 75, leaving remainders of 2 and 5 respectively, we can follow these steps:

### Step 1: Set up the equations
Let the largest number be \( x \). According to the problem:
- When 92 is divided by \( x \), it leaves a remainder of 2.
- When 75 is divided by \( x \), it leaves a remainder of 5.

This can be expressed mathematically as:
1. \( 92 - 2 = kx \) for some integer \( k \) (which means \( 90 \) is divisible by \( x \))
2. \( 75 - 5 = mx \) for some integer \( m \) (which means \( 70 \) is divisible by \( x \))

### Step 2: Find the numbers to be divided
From the equations above, we find:
- \( x \) must divide \( 90 \) (from \( 92 - 2 = 90 \))
- \( x \) must divide \( 70 \) (from \( 75 - 5 = 70 \))

### Step 3: Find the common divisors
Now, we need to find the greatest common divisor (GCD) of \( 90 \) and \( 70 \).

### Step 4: Factor the numbers
- The prime factorization of \( 90 \) is \( 2 \times 3^2 \times 5 \).
- The prime factorization of \( 70 \) is \( 2 \times 5 \times 7 \).

### Step 5: Identify common factors
The common factors of \( 90 \) and \( 70 \) are \( 2 \) and \( 5 \).

### Step 6: Calculate the GCD
The GCD is found by taking the lowest power of all common prime factors:
- For \( 2 \): \( 2^1 \)
- For \( 5 \): \( 5^1 \)

Thus, the GCD is:
\[ 2^1 \times 5^1 = 10 \]

### Step 7: Conclusion
The largest number \( x \) that divides both \( 90 \) and \( 70 \) is \( 10 \).

### Final Answer
The largest number that divides 92 and 75, leaving remainders of 2 and 5 respectively, is **10**.

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