The greatest number, which when subtracted from 5834, gives a number exactly…

To find the greatest number which, when subtracted from 5834, leaves a number exactly divisible by 20, 28, 32, and 35, we need to work with the Least Common Multiple (LCM) of these divisors.

Step-by-Step Calculation
1. Find the LCM of the divisors (20, 28, 32, 35):

Prime factorize each:

20 = 2^2 * 5

28 = 2^2 * 7

32 = 2^5

35 = 5 * 7

The LCM is the product of the highest powers of all prime factors present:

Factors: 2^5, 5^1, 7^1

LCM = 32 * 5 * 7 = 1120.

2. Analyze the remainder:
We want the result of (5834 - X) to be a multiple of 1120.

Divide 5834 by the LCM (1120):

5834 / 1120 = 5 with a remainder of 234.

This means 5834 = (1120 * 5) + 234.

3. Determine the number to subtract:
To make 5834 divisible by 1120, we must subtract the remainder (234) so that we are left with the multiple 5600.

5834 - 234 = 5600 (which is 1120 * 5, exactly divisible).

The "greatest number to subtract" to achieve a result divisible by the LCM is the remainder itself: 234.