Random Puzzles 

Workings

Hint
Try working backwards.

Answer
148.

Reasoning
A number that is divisible by 3 has digits that add up to a number that is also divisible by 3 (for example, 123 is divisible by 3).

So we can look for the first number where 4 + 4 + 4 + 4 + 4 + … is divisible by 3, which is 4 + 4 + 4, and 444 ÷ 3 = 148.

Alternative Reasoning
Let's try the possible answers in order:

  4 ÷ 3 = 1.3333
 44 ÷ 3 = 14.6666
444 ÷ 3 = 148

Double-Checking
148 x 3 = 444.

Workings

Hints
Five of the missing first letters are: F, P, T, C, B.

Answers
--pay = repay --lad = salad --ior = prior --eue = queue --rva = larva --day = today --voc = havoc --yss = abyss --gun = begun --pty = empty --til = until --cay = decay --uke = fluke --uze = gauze --noe = canoe

Workings

Hint
Look down?

Answer
O.

Every other letter from the top row of a standard keyboard.

Workings

Hint
Does it start with January?

Answer
28.

The number of days in each month starting from October. The answer could also be 29 for a leap year.

The answer can't be 30, as the question specifically says that the answer is not 30, so it can't be the sequence starting with May.