What number must be subtracted from each of the numbers 53, 21, 41, 17 so that the remainders are in proportion?

Options

A. 1
B. 3
C. 5
D. none of these

Answer

To solve the problem of finding the number that must be subtracted from each of the numbers 53, 21, 41, and 17 so that the remainders are in proportion, we can follow these steps:

### Step 1: Define the variable
Let \( x \) be the number that we need to subtract from each of the given numbers.

### Step 2: Write the expressions for the remainders
After subtracting \( x \) from each number, the remainders will be:
- From 53: \( 53 - x \)
- From 21: \( 21 - x \)
- From 41: \( 41 - x \)
- From 17: \( 17 - x \)

### Step 3: Set up the proportion
We need to set up the proportion of the remainders:
\[ \frac{53 - x}{21 - x} = \frac{41 - x}{17 - x} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ (53 - x)(17 - x) = (21 - x)(41 - x) \]

### Step 5: Expand both sides
Expanding both sides:
- Left side:
\[ 53 \cdot 17 - 53x - 17x + x^2 = 901 - 70x + x^2 \]
- Right side:
\[ 21 \cdot 41 - 21x - 41x + x^2 = 861 - 62x + x^2 \]

### Step 6: Set the equation
Now we have:
\[ 901 - 70x + x^2 = 861 - 62x + x^2 \]

### Step 7: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ 901 - 70x = 861 - 62x \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 901 - 861 = 70x - 62x \]
\[ 40 = 8x \]

### Step 9: Solve for \( x \)
Dividing both sides by 8:
\[ x = 5 \]

### Conclusion
The number that must be subtracted from each of the numbers 53, 21, 41, and 17 to make the remainders proportional is \( \boxed{5} \).

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