Let's find the greatest number which divides 285 and 1245 and leaves a remainder of 5 in each case step by step:
Step 1: First, we find the numbers that would result if each were divided completely by the unknown greatest number (that is, excluding the remainder of 5 in each case). We do this by subtracting the common remainder from each number.
For 285, since it leaves a remainder of 5 when divided by the unknown greatest number, the completely divisible part would be \( 285 - 5 = 280 \).
For 1245, the completely divisible part would be \( 1245 - 5 = 1240 \).
Step 2: Now, we need to find the greatest common divisor (GCD) of 280 and 1240 because the GCD will give us the largest number that can exactly divide both 280 and 1240.
Step 3: To find the GCD of 280 and 1240, we can use the Euclidean algorithm, which involves dividing the larger number by the smaller number and then using the remainder as the new divisor until the remainder becomes zero. The divisor that results in a zero remainder is the GCD.
We can start by dividing 1240 by 280:
\( 1240 ÷ 280 = 4 \) with a remainder of 0, which means the GCD is 280.
However, in our case, we already know that 280 is a multiple of 1240 (since 1240 is 4 times 280), so the GCD is indeed 280. But, we are not looking for a number that divides 280 and 1240; we are looking for a number that divides 285 and 1245 with a remainder 5.
Knowing that 280 is the GCD of the two complete divisible parts suggests that our solution cannot be larger than 280, since both 285 and 1245 are not divisible by 280.
Let's check if there are any common factors between 280 and the number 5 (the remainder), which would indicate that we need to find a smaller number than 280 that can also account for the remainders when dividing both 285 and 1245.
Since 5 is a prime number, there are no common factors between 280 and 5 other than 1. Therefore, our GCD remains 280.
Step 4: Finally, because we are interested in the greatest number that leaves the same remainder of 5 when dividing both original numbers (285 and 1245), we realize that this greatest number must divide both 280 and 1240, which is 280 itself.
So, the greatest number that divides 285 and 1245, leaving a remainder of 5 in each case, is actually 280. Therefore, the answer provided initially (40) is not correct; the correct answer is 280.
Note: This solution assumes that the task is to find the greatest number that divides both 285 and 1245, not the numbers obtained from subtracting the remainder. If we were to only consider the complete divisible portion from both numbers with a common remainder of 5 excluded, then yes, the greatest number (the GCD) would be 280. But if the question is to consider the remainder, then the solution given would typically represent the highest common factor of two reduced numbers (280 and 1240), which is in this case 280.
