Tournament of Towns in Toronto

Practice session January 18, 2014. Divisibility. Modular Arithmetics
  1. Anna thought of a number and added to it the sum of its digits. Ben also thought of a number and added to it the sum of its digits. It occurred that Anna' and Ben' results were the same. Can it be that Anna and Ben thought of the different numbers?
  2. Alex and Ben found two bags with 11-tugrics bills. At Mr. Fancy's restaurant Alex ordered 3 cups of tea, four donuts and 5 pretzels while Ben ordered 9 cups of tea, one donut and 4 pretzels. Each item, tea, donut and pretzel costs integer number of tugrics. It occurred that Alex's order summed up to integer numbers of bills. Prove that Ben's order summed up to integer numbers of bills as well.
  3. Find the last digit of product of all odd three digit numbers.
  4. What is the last digit of $3^{2013}$?
  5. Find the smallest multiple of 45 which is composed of 1-s and 0-s only.
  6. Determine the smallest number which does not divide $31!$
  7. Determine the maximal power of $3$ that divides $3\times33\times 333\times \ldots \times 3,333,333,333$.
  8. A teacher gave his pupils a homework to exercise during Christmas break. Starting from number (20+14)! one must obtain a sequence of numbers, each number is equal to the sum of digits of the previous number. The sequence must terminate when two terms coincide. What is the last term?
  9. Alex and Max looked at a $100\times100$ table filled with with non zero digits. Alex noticed that each of the 100-digit numbers formed by the rows is divisible by 9 while Max observed that each of 100-digit numbers but one formed by the columns is divisible by 9. Could both the boys be right?
  10. The number $n$ is three times more than sum of its digits. Prove that $n$ is divisible by 27.
  11. Anna wrote a four digit number. Her brother Ben wrote it backwards. Then they calculated the sum of these two numbers. Anna noticed that it was divisible by 11. For which numbers could this be the case?
  12. A number is composed of one hundred of the 0s, one hundred of the 1s, and one hundred of the 2s. Can it be a perfect square?
  13. Prove that the number $1234567891011\ldots20182019$ is not a perfect square.
  14. A certain number leaves a remainder 1 when divided by 2 while it leaves a remainder 2 when divided by 3. What is the remainder when the number is divided by 6?
  15. A grandfather brought apples from his garden (less than 100). His grandchildren arranged them into piles of two apples, and noticed that one apple was left over. Then, they arranged apples into piles of three, four, five and six apples and each time, one apple was left over. How many apples did the grandfather bring from his garden?
  16. A grandfather brought apples from his garden (less than 100). His grandchildren played with apples. Arranging apples into piles of two, then three, then four, then five and six the children noticed that each time respectively one, two, three, four, five apples were left over. How many apples did the grandfather bring from his garden?
  17. Prove that $k^3-k$ is divisible by 6 for every integer $k$.
  18. Prove that out of 10 positive integers, none of which is divisible by 10, one can select
    1. two numbers whose difference is divisible by 10;
    2. several numbers whose sum is divisible by 10.
  19. Prove that among any 51 integers, one can find two numbers such that their squares have the same remainders when divided by 100.
  20. Take the number $8^{2014}$ and calculate the sum of its digits to obtain a new number. Repeat this operation until you are left with one digit number. What is this number?
  21. The sum of squares of two integers is divisible by 3. Prove that each integer is divisible by 3.
  22. Find remainders of the following numbers when divided by $n$:
    1. $900364+45812+ 55431$, $n= 9$;
    2. $2014 \times 2013$, $n=11$;
    3. 38, $38\times38$, $38^3$, $n=3$;
    4. $7^{2014}$, $n=8$.
  23. Prove that the following numbers
    1. $30^{97}+63 ^{79}$ is divisible by 31;
    2. $1^{17}+2^{17}+\dots+16^{17}$ is divisible by 17.
  24. What is the smallest term one must add to expression $(n^2+1)^{1000}\times (n^3-1)^ {1001}$ to get the result divisible by $n$?
  25. Let $p$ and $q$ be prime numbers. How many divisors have the following numbers
    1. $pq$;
    2. $p^2q^2$;
    3. $p^{m}q^{n}$.
  26. For $30!$ determine the number of its divisors.
  27. Determine the smallest number $n$ such that $n!$ is divided by $10^{10}$.
Problems for whole family
  1. Serge and Paul caught 7 and 5 fish respectively while Nick caught none. They cooked soup of these fish and shared it equally. As his share, Nick gave Serge and Paul 12 fishhooks. How should Serge and Paul divide these fishhooks?
  2. Nicolas with his son and Peter with his son were fishing. Nicolas and his son caught the same number of fish while Peter caught three times more than his son. All of them together caught 25 fish. How many fish did Nicolas catch?


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