40. Write the number 1980 in words and count the total number of letters in these words. Write the number expressing the count of letters in words and count the letters in them again, then repeat this operation 10 more times and write down all 13 numbers in sequence.

a) Identify the pattern in writing this sequence starting from the fifth number.

b) Prove that no matter what number the sequence starts with, a certain group of numbers will begin to repeat in a certain order starting from a certain position.

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40. "One thousand nine hundred eighty" has 26 letters, "twenty-six" has 13 letters, "thirteen" has 10 letters, "ten" has 6 letters, "six" has 5 letters, "five" has 4 letters, "four" has 6 letters, "six" has 5 letters, "five" has 4 letters. We have:

$1980, 26, 13, 10, 6, 5, 4, 6, 5, 4, 6, 5, 4$.

a) Starting from the fifth number, i.e., the number 6, the group of numbers $6, 5, 4$ repeats.

b) No matter what the first number in the sequence is, the number of letters in the words expressing its name will be a certain number. For numbers with two or more digits, each subsequent number in the sequence is smaller than the previous one. Therefore, at some point in the sequence, a single-digit number will appear. The smallest number that can appear is 3, since the smallest number of letters in the names of numbers is three. (For example, the number 100.) But if the number 3 appears, all subsequent numbers in the sequence will be threes. Thus, one number in the sequence will repeat. If the first single-digit number to appear is 4 (four), then the next will be 6 (six), followed by 5 (five), then 4, i.e., the numbers $4, 6, 5$ form a repeating cycle.

If the first single-digit number to appear is 5 (five), then the next will be 4, followed by 6, then 5, and so on. Thus, the cycle $5, 4, 6$ repeats. If the first single-digit number to appear is 6, then the next will be 5, followed by 4, 6, and so on, i.e., the cycle $6, 5, 4$ repeats. If the first single-digit number to appear is 7 (seven), then the next will be 4, and as seen from the previous, the cycle $4, 6, 5$ will appear. If the first single-digit number to appear is 8 (eight), then the cycle $6, 5, 4$ will appear. Finally, if the first single-digit number to appear is 9 (nine), then the cycle $6, 5, 4$ will appear again. Thus, either the number 3 will repeat, or one of the triplets $(4, 6, 5), (5, 4, 6), (6, 5, 4)$.

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Arithmetic operations on natural numbers and zero and their properties