When you divide 2149 by n, the remainder is 4. When you divide 1578 by n the remainder is 5. What is the least possible natural number n? and other question

When you divide 12,650 by n , the remainder is 2. When you divide 5,800 by n the remainder is 3. What is the least possible natural number n?

Bonus If you want answer these 2 other questions

n is a natural number. Find the greatest possible value of the GCD of
(2n+25) and (n+15)
Also answer
(2n+25) and (n+15)

First Q: The smallest value for which n works is 11.

Second Q: Smallest here is 17.

Third Q: I don't know this one, sorry!

Fourth Q: n is -10.

Step-by-step explanation:

For the first two questions, find the smallest common factor between the two numbers (subtract the remainder).

Q1:

First, subtract the remainders from the numbers (2149 - 4 = 2145; 1578 - 5 = 1573)

Then, list the factors

2145: 1, 3, 5, 11, 13, 15, 33, 39, 55, 65, 143, 165, 195, 429, 715, 2145

1573: 1, 11, 13, 121, 143, 1573

These two numbers share 11 and 13 in common. 11 is the smaller of these, so it is the smallest value n can work.

Q2:

Repeat the steps from Q1.

12650 - 2 = 12648; 5800 - 3 = 5797

12648: 1, 2, 3, 4, 6, 8, 12, 17, 24, 31, 34, 51, 62, 68, 93, 102, 124, 136, 186, 204, 248, 372, 408, 527, 744, 1054, 1581, 2108, 3162, 4216, 6324, 12648

5797: 1, 11, 13, 17, 31, 187, 341, 527, 5797

These two share 17, 31, and 527 in common. 17 is the smallest of these, so it is the smallest value n can work.

Q4 (I don't know how to do Q3 yet):

2n + 25 = n + 15

Subtract n from both sides

n + 25 = 15

n = 15 - 25

Subtract 25 from 15 to get -10

n = -10

Again, I don't know how to do Question 3 yet. Sorry for the inconvenience! Anyhow I hope the rest of this was helpful!

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step-by-step explanation: