DIVISIBILITY AND PRIME NUMBERS

MR. T’S SONS.
The age of each of Mr. Triangle’s three sons is an integer. The sum of these integers equals 12, and their arithmetic product is 30. How old is each of Mr. Triangle’s sons?

A ONE HUNDRED-HEADED DRAGON.
Once upon a time, there lived a fierce dragon, which had a hundred heads. With a stroke of his sword, the knight could cut off one, seven or 11 heads, but if at least one head remained uncut, immediately after the sword stroke, there grew back four, one, or five heads, respectively. Was the knight able to kill the dragon, then? What would be the answer if the dragon had initially had 99 heads?
Remember: The dragon dies if after the sword stroke he has no more heads

A COLUMN OF PLASTIC TROOPS.
Bart has an army of plastic soldiers. When he tried to form with his soldiers a column of fours, in the last row remained only three figures. When Bart formed a column of threes, the last row consisted of only two soldiers. How many soldiers will he have in the last row if he forms a column of sixes?

SPECIAL NATURAL NUMBERS
Can you find 10 different natural numbers whose sum is a number divisible
by each of these numbers?
Clue: You should start your attempt to solve this problem with three natural
numbers.

NOSTRADAMUS AND HIS PROPHECY
According to Nostradamus, a famous French apothecary and a famous seer
(1503-1566), exceptional are those years which written in the decimal
system have the form abcd and comply with ab + cd = bc, where ab, cd and
bc denote two-digit numbers which are also written in the decimal system.
It is assumed at the same time that if c = 0, then 0d denotes a single-digit
number d. For instance, the year 1208 was exceptional because 12 + 08 = 20. Which nearest year after 2006 will be exceptional?

THE MAGNIFICENT SEVEN
Seven integers have been chosen such that the sum of any two numbers is
divisible by 7. How many numbers of the selected set are divisible by 7?