The factors of a number are all positive integers that divide evenly into the number. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.

To simplify this fraction, first find the greatest common factor between them. The factors of 20 are [1, 2, 4, 5, 10, 20] and the factors of 68 are [1, 2, 4, 17, 34, 68]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{20}{68} \) = \( \frac{\frac{20}{4}}{\frac{68}{4}} \) = \( \frac{5}{17} \)

To divide fractions, invert the second fraction and then multiply:

\( \frac{2}{7} \) ÷ \( \frac{4}{5} \) = \( \frac{2}{7} \) x \( \frac{5}{4} \)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{2}{7} \) x \( \frac{5}{4} \) = \( \frac{2 x 5}{7 x 4} \) = \( \frac{10}{28} \) = \(\frac{5}{14}\)

Calculate the number of vans needed by dividing the number of people that need transported by the capacity of one van:

vans = \( \frac{67}{7} \) = 9\(\frac{4}{7}\)

So, it will take 9 full vans and one partially full van to transport the entire team making a total of 10 vans.