Q1.
Are there other expressions that generate numbers that are 3 more than a multiple of 5?
Answer
Infinitely many. Any expression 5k + c works, provided c itself leaves a remainder of 3 on division by 5.
5k + 8 (8 = 5 + 3)
5k + 13 (13 = 10 + 3)
5k – 7 (– 7 = – 10 + 3)
5k – 12 (– 12 = – 15 + 3)
5k + 13 (13 = 10 + 3)
5k – 7 (– 7 = – 10 + 3)
5k – 12 (– 12 = – 15 + 3)
All of them run through the same list 3, 8, 13, 18, 23, … — only the starting value of k shifts.
| Expression | k = 0 | k = 1 | k = 2 | k = 3 |
|---|---|---|---|---|
| 5k + 3 | 3 | 8 | 13 | 18 |
| 5k + 8 | 8 | 13 | 18 | 23 |
| 5k – 7 | – 7 | – 2 | 3 | 8 |
Tip: what does not work is changing the 5. An expression like 10k + 3 gives only 3, 13, 23, … — correct numbers, but it misses 8, 18, 28. To capture all of them the coefficient must be exactly 5.