Arithmetic

Arithmetic is the fundamental branch of mathematics dealing with numbers and the basic operations: addition, subtraction, multiplication, and division. Questions involve performing these operations, understanding number properties (like integers, fractions, decimals), and solving related word problems.

• Question

  Is it possible for the sum of three natural numbers to be divisible by each of them?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • problems.ru web site

• Question

  All the even numbers from `12` to `34` are written on the board without spaces. As a result, the following number was obtained:

  `121416182022242628303234`

  Is this number divisible by `24`?

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  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 2, 4, and 8 Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9
  Sources:
  • problems.ru web site

• Question

  I went to the market to buy vegetables. I paid for tomatoes half of the money I had plus one shekel. After that, I paid for cucumbers half of the remaining money plus one shekel, and then I paid for onions half of the remaining money plus one shekel, and that used up all the money I had.

  What is the amount of money I came to the market with?

  Topics:

  Arithmetic Algebra -> Equations Algebra -> Word Problems -> Solving Word Problems "From the End" / Working Backwards
  Sources:
  • problems.ru web site

• Question

  The following numbers are written on the board:

  `1,2^1,2^2,2^3,2^4,2^5`

  In one operation, you are allowed to erase two numbers written on the board and write their (non-negative) difference in their place.

  Is it possible to reach a state, through such operations, where only the number `15` is written on the board?

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  Topics:

  Arithmetic Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences
  Sources:
  • problems.ru web site

• Question

  The number `100` is written on the board. Find a digit that satisfies the following condition:

  If we add it to the notation of the number written on the board once to the left and once to the right, we get a number that is divisible by `12`.

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  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 2, 4, and 8 Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Prime Numbers -> Prime Factorization
  Sources:
  • problems.ru web site

• Clock Tower

  A bell in a clock tower rings `3` times in `12` seconds. How long does it take to ring `6` times?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Number Theory -> Division
  Sources:
  • problems.ru web site

• Question

  Which number is larger:

  `2^300` or `3^200`?

  Topics:

  Arithmetic Algebra -> Inequalities

• The Restless Fly

  Cities A and B are 300 kilometers apart. Two cyclists start simultaneously from A and B, heading towards each other. Their speeds are constant and equal to `30` km/h and `20` km/h, respectively. At the same moment, a fly departs from city A, flying at a speed of `100` km/h. The fly overtakes the first cyclist and flies until it meets the cyclist who left from city B. The moment the fly meets the cyclist, it immediately turns around and flies back until it meets the first cyclist again, and then it turns around again, and so on, until the cyclists meet. How many kilometers did the fly travel?

  Topics:

  Arithmetic Algebra -> Equations Logic -> Reasoning / Logic Algebra -> Word Problems -> Motion Problems

• Question

  What is the smallest four-digit number whose first digit is `8`, the number is divisible by `3`, and all its digits are different?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Division

• Question

  At `12:00`, the two hands of a clock coincide. In exactly how much time will they coincide again? How many times within a 24-hour period will they coincide?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Word Problems -> Motion Problems