Read the two schedules as two separate filters, because the two columns ask completely different things. The first vendor appears in weeks 1, 6, 11, 16, and so on, which are the weeks one more than a multiple of 5. The second vendor appears in weeks 1, 8, 15, 22, and so on, which are the weeks one more than a multiple of 7. The first selection wants the weeks that pass BOTH filters; the second selection wants a week that passes only the second filter. Solving one tells you nothing about the other, so each needs its own scan.
First column. Both vendors appear together exactly when the week is one more than a multiple of both 5 and 7, that is, one more than a multiple of 35. The first such week after week 1 is 1 + 35 = 36, the entry in the sixth row. A common slip is to report 35, the gap between coincidence weeks, instead of the week number itself, which lands on the fifth row, 35. Another slip picks 6, the first week the first vendor alone reappears, or 8, the first week the second vendor alone reappears; neither is a week both attend.
Second column. Walk the second vendor's weeks after week 1: 8, 15, 22, and so on. Test the first one, week 8, against the first vendor: 8 is not one more than a multiple of 5, since 8 leaves a remainder of 3 when divided by 5, so the first vendor is absent in week 8. Week 8 is therefore the answer, the entry in the second row. The value 15 is the second week where only the second vendor appears, so picking it means skipping the first qualifying week, 8. The value 6 is the first week where only the first vendor appears, which is the same idea with the vendors swapped. The value 13 comes from advancing both schedules once from week 1, that is 1 + 5 + 7, but 13 is not even one of the second vendor's weeks. The value 36 is the both-together week from the first selection, chosen by a solver who answered the wrong column.
The answer to the first column is 36 and the answer to the second column is 8.