To find the greatest number that divides 418, 1354, 2146, and 2338, each leaving a remainder of 10, we'll follow these steps:
1. Adjust the Given Numbers: Since each number leaves a remainder of 10 when divided by the required divisor, subtract 10 from each number to get:
- 418 - 10 = 408
- 1354 - 10 = 1344
- 2146 - 10 = 2136
- 2338 - 10 = 2328
2. Find the Greatest Common Divisor (GCD): The next step is to determine the GCD of these adjusted numbers. The GCD is the largest number that divides all the given numbers without leaving a remainder.
Let's break this down step-by-step:
- Calculate the GCD of the first two numbers: 408 and 1344.
- Next, find the GCD of that result with the third number: 2136.
- Lastly, determine the GCD of the resulting value with the fourth number: 2328.
Here are the steps in detail:
- Compute the GCD of 408 and 1344: The result is 24.
- Compute the GCD of 24 and 2136: The result is still 24, since any number that divides 24 also divides its multiples or larger numbers formed by multiplying with integers.
- Compute the GCD of 24 and 2328: The result remains 24 for the same reasons mentioned above.
Therefore, the greatest number which divides 418, 1354, 2146, and 2338 exactly, leaving a remainder of 10 in each case, is 24.
