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.

• Continue the Sequence

  `2, 6, 12, 20, 30…`

  a. What is the next number (the sixth) in the sequence?

  b. What is the `2016`th number in the sequence?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Sequences -> Complete/Continue the Sequence
  Sources:
  • problems.ru web site

• Question

  It is known that `a = b+1`. Is it possible that `a^4=b^4`?

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  Topics:

  Arithmetic Algebra -> Equations
  Sources:
  • problems.ru web site

• Question

  Can you find `35` integers whose average is equal to `6.35`?

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  Topics:

  Arithmetic Combinatorics -> Double Counting Algebra -> Inequalities -> Averages / Means Proof and Example -> Proof by Contradiction
  Sources:
  • problems.ru web site

• Question

  Which of these two numbers is larger:

  `A=2011*20122012*201320132013 `  or  `B=2013*20112011*201220122012` ?

  Topics:

  Arithmetic Algebra -> Algebraic Techniques
  Sources:
  • problems.ru web site

• Question

  Is it possible to make change for a `25` lira note using `10` coins worth `1`, `3`, and `5` lira?

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  Topics:

  Arithmetic Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd)
  Sources:
  • problems.ru web site

• Movie Buffs

  Avi, Beni, and Gili love movies. Avi goes to the movies every `3` days, Beni – every `5` days, and Gili goes every `7` days. Today, they all went to the movies together. In how many days can this happen again?

  Topics:

  Arithmetic Algebra -> Word Problems Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM)
  Sources:
  • problems.ru web site

• Question

  The "Sweet Math" candies are sold in boxes of `12` units, and the "Geometry with Nuts" candies – in boxes of `15` units.

  What is the minimum number of boxes that must be purchased so that there are equal quantities of candies of both types?

  Topics:

  Arithmetic Algebra -> Word Problems Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Minimum and Maximum Problems / Optimization Problems
  Sources:
  • problems.ru web site

• Question

  The number `458` is written on the board. In each single step, you are allowed to either multiply the number written on the board by `2`, or erase its last digit.

  Is it possible to obtain the number `14` using these operations?

  Topics:

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

• Question

  How many two-digit numbers are there such that the tens digit is greater than the units digit?

  Topics:

  Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Ali Baba and the Forty Thieves

  Ali Baba wrote the number `17` on a piece of paper. The forty thieves pass the paper to each other, and each one either adds `1` to the existing number, or subtracts `1`, until each of them has done so once, and then they return the paper to Ali Baba.

  Is it possible that the number now written on the paper is `40`?

  Topics:

  Arithmetic Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site