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

Question

Accepted Answer

To determine the least number that must be subtracted from each of the numbers 14, 17, 34, and 42 so that the remainders are proportional, we need to find a common factor that makes the ratios of the remainders equal.

Let's denote the number to be subtracted as $ x $. After subtracting $ x $ from each number, the remainders will be $ 14-x $, $ 17-x $, $ 34-x $, and $ 42-x $. We want these remainders to be proportional, meaning:

$$
\frac{14-x}{17-x} = \frac{34-x}{42-x}
$$

Cross-multiplying gives:

$$
(14-x)(42-x) = (17-x)(34-x)
$$

Expanding both sides:

$$
588 - 56x + x^2 = 578 - 51x + x^2
$$

Simplifying the equation by canceling $ x^2 $ and rearranging terms:

$$
588 - 578 = 56x - 51x
$$

$$
10 = 5x
$$

Solving for $ x $:

$$
x = 2
$$

Thus, the least number that must be subtracted from each of the numbers is 2, which makes the remainders proportional.

**Explanation for why other options are incorrect:**

- **Option 2 (3):** If we subtract 3 from each number, the remainders are 11, 14, 31, and 39. The ratios $\frac{11}{14}$ and $\frac{31}{39}$ are not equal, so the remainders are not proportional.

- **Option 3 (5):** If we subtract 5 from each number, the remainders are 9, 12, 29, and 37. The ratios $\frac{9}{12}$ and $\frac{29}{37}$ are not equal, so the remainders are not proportional.

- **Option 4 (7):** If we subtract 7 from each number, the remainders are 7, 10, 27, and 35. The ratios $\frac{7}{10}$ and $\frac{27}{35}$ are not equal, so the remainders are not proportional.

Therefore, Option 1 is the correct answer, as subtracting 2 from each number results in proportional remainders.