Question

Find the greatest number which divides 350 and 511 leaving the remainder 5 in each case​

Answer 1

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The largest number which can be divided from both 129 and 545 with respective remainders of 3 and 5 is 18.

Explanation: It isn’t clear if we want to divide by 129 and 545 or from 129 and 545. We can deduce, though, that we want to divide from those numbers, as the amount of numbers which can divide by these numbers is infinite. We want to find the greatest number that can be divided from 129 and 545 to produce remainders of 3 and 5, respectively. This can be done by figuring what numbers we can divide from without the remainders then finding the Greatest Common Factor (GCF.)

The first step can be used to make this question easier. It is to subtract 3 from 129 and 5 from 545. This is permissible because 3 and 5 will be our remainders, and later steps are best done when division will exclude remainders. Therefore, we are left with 126 and 540, respectively. With these numbers, we can move onto the next step.

The next step is to find the GCF of these two numbers. To do so, we must list the factors of each of these two numbers and determine which is the greatest that appears on both lists. Let us start by listing the factors of 126:

1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126.

Next, let us list the factors of 540:

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540.

Next, we look through each of the numbers and find the largest which appears in both of the lists. We can do so by combining the lists of factors then going through all of them in the big list to see which appear in both, but that would be tedious. Therefore, I will just say which number it is: 18.

The largest number which can be divided by both 126 and 540 is 18. Therefore, dividing it from 129 and 545 will produce respective remainders of 3 and 5.

Checking our work: Let us make sure 18 works by dividing it from 129 and 545 with remainders. After that, we can check by multiplying 18 by the whole number results then adding the remainders.

129 / 18 = 7 R 3

(18 * 7) + 3 = 129

545 / 18 = 30 R 5

(18 * 30) + 5 = 545

Our multiplication/addition problems came out to 129 and 545, so 18 can indeed be divided from those numbers with the respective remainders of 3 and 5.