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Intellectual quiz.

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Offline Fireball
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Posts: 886
#23
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Vasya, a poor student, tore three sheets out of his diary. He tore some of them (maybe all) into three pieces or five pieces. He tore some of the resulting pieces of paper into three or five pieces again, and so on. Could Vasya, by doing this, obtain exactly 2002 pieces of paper?

Answers "yes" and "no" are not accepted; describe the solution.
Initially, Vasya had 3 pieces of paper – an odd number. Each tear into 3 or 5 pieces increases the total number of fragments of the unfortunate diary by 2 or 4 – an even number. That is, by acting in this way, Vasya can only obtain an odd number of pieces, and 2002 is an even number. Therefore, the answer is no, he could not.
And, at Fireball's request, a more difficult problem, 8th grade, again mathematics:

Someone, when asked what his ticket number was, answered as follows: "If all 6 two-digit numbers that can be formed from the digits of the number are added together, then half of the resulting sum will be exactly my ticket number." Determine the ticket number.
Answer: 198. The problem boils down to solving the equation 89a = b + 10c, where a, b, c are integers from 0 to 9, but a is not equal to zero.
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