For certain values of the whole number x, the fraction
| 9x + 7 |
| 6x + 9 |
will not be in lowest terms (simplest form). Explain and give examples.
If you set this up on a spreadsheet, you’ll find that x = 5, 18, 31, 44, 57, etc., all give fractions that are not in lowest terms.
We observe that these x‘s are of the form 13k + 5, for k = 0, 1, 2, 3, …
So let’s see:
| 9x + 7 | = | 9(13k + 5) + 7 | = | 117k + 45 + 7 | = | 117k + 52 | = |
| 6x + 9 | 6(13k + 5) + 9 | 78k + 30 + 9 | 78k + 39 |
| 9 · 13k + 4 · 13 | = | 13(9k + 4) | . |
| 6 · 13k + 3 · 13 | 13(6k + 3) |
My, my.