Medium Puzzles 

Workings

Hint
Try writing the numbers as words.

Answer
21.

These are numbers ending in a vowel (AEIOU).

Workings

Hint
The most common word begins with the letter V.

Answers
Common: vacuum, vacuumed, vacuuming, vacuums.

Less common: continuum, continuums, muumuu, muumuus.

Much less common: carduus, carduuses, continuua, duumvir, duumviral, duumvirate, duumvirates, duumviri, duumvirs, individuum, individuums, lituus, lituuses, menstruum, menstruums, mutuum, mutuums, paramenstruum, paramenstruums, residuum, residuums, squush, squushed, squushes, squushier, squushiest, squushing, squushy, triduum, triduums, ultravacuum, ultravacuums, weltanschauung, weltanschauungen, weltanschauungs.

Workings

Hint
The cube root of the given number is a good starting point.

Answer
71, 73, 79.

Reasoning
The cube root of 409,457 is around 74.

The closest consecutive prime numbers to 74 are: 61, 67, 71, 73, 79, 83, 89, 97.

Using a calculator, it is quickly seen that the answer can only be found using 71, 73 and 79.

Workings

Hint
The answer involves months of the year.

Answer
6.

The number of letters each month has, starting with January, therefore, August has 6 letters.