1. Prove that sure that a) 1/3 = 0,(3); b) 1/6 = 0.1(6); c) 7/30 = 0.2(3); d) 7/11 = 0.(63).

2. Find the hundredth digit after the decimal point of 1/7.

3. Divide number 1 by

a) 9;

b) 99;

c) 9,999.

d) Prove the general rule: 1/99...9 = 0,(0...01). (In the denominator there are n nines, in brackets there are (n – 1) zeros.)

e) A purely periodic proper fraction is equal to an ordinary fraction in which the numerator is the period, and the denominator is the number 10r – 1 = 9...9 (r nines), where r is the length of the period. Prove it.

4. Convert the numbers to decimal fractions: a) 23/99; b) 1234/999999.

5. How many multiples of 13 are there among the first hundred numbers of the sequence 1, 11, 111, 1111,...?

6. If a number of the form 11...11 is a multiple of 7, then it is a multiple of 11, 13, and 15873. Prove this.

7. The first digit of a k-digit number that is a multiple of 13 was erased and written behind the last digit of that number. For what k is the resulting number a multiple of 13? (For example, from the multiples of 13 numbers 503,906 and 7969, we thus obtain the numbers 39,065 and 9,697, the first of which is a multiple of 13, and the second is not.)

8. The number ends in 2. If this digit is moved to the beginning of the number, it will double. Find the smallest such number.

9. What digit ends in 3377 + 7733?

10. Find the last two digits of 22000.

11. What is the last digit of 9999999999?

12. A five-digit number is divisible by 41. Prove that if its digits are cyclically rearranged, then the resulting number will also be divisible by 41. (For example, knowing that 93,767 is divisible by 41, one can say that 37,679 is divisible by 41.)

13.

a) Consider the numbers: 1, 11, 111, 1111,... Prove that among them there are two numbers whose difference is divisible by 196,673.
b) Prove that there is a number written only in units and a multiple of the number 196,673.
c) For any natural number a that is not divisible by either 2 or 5, there is a natural number b such that the product ab is written in the decimal number system using only units. Prove it.

14. If natural numbers a and m are coprime, then there exists a natural number n such that an – 1 is divisible by m. Prove it.