What least number must be subtracted from each of the numbers 14, 17, 34 and 42 so that the remainders may be proportional ?

Options

A. 0
B. 1
C. 2
D. 7

Answer

To solve the problem step by step, we need to find the least number that must be subtracted from each of the numbers 14, 17, 34, and 42 so that the remainders are proportional.

### Step 1: Understand the concept of proportion
To say that four numbers \( a, b, c, d \) are in proportion means that the ratio of the first two numbers is equal to the ratio of the last two numbers. Mathematically, this can be expressed as:
\[ \frac{a}{b} = \frac{c}{d} \]

### Step 2: Set up the equation
Let \( x \) be the least number that we need to subtract from each of the given numbers. After subtracting \( x \), the new numbers will be:
- \( 14 - x \)
- \( 17 - x \)
- \( 34 - x \)
- \( 42 - x \)

We want these numbers to be in proportion:
\[ \frac{14 - x}{17 - x} = \frac{34 - x}{42 - x} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ (14 - x)(42 - x) = (17 - x)(34 - x) \]

### Step 4: Expand both sides
Now we will expand both sides of the equation:
- Left side:
\[ 14 \cdot 42 - 14x - 42x + x^2 = 588 - 56x + x^2 \]
- Right side:
\[ 17 \cdot 34 - 17x - 34x + x^2 = 578 - 51x + x^2 \]

### Step 5: Set the equation to zero
Now we set the equation:
\[ 588 - 56x + x^2 = 578 - 51x + x^2 \]
Subtract \( x^2 \) from both sides:
\[ 588 - 56x = 578 - 51x \]

### Step 6: Simplify the equation
Rearranging gives:
\[ 588 - 578 = 56x - 51x \]
\[ 10 = 5x \]

### Step 7: Solve for \( x \)
Dividing both sides by 5:
\[ x = 2 \]

### Conclusion
The least number that must be subtracted from each of the numbers 14, 17, 34, and 42 so that the remainders are proportional is \( \boxed{2} \).