Number Theory, Modular Arithmetic / Remainder Arithmetic, Divisibility Rules, Divisibility Rules by 5 and 25

This covers the divisibility rules for 5 (if the last digit is 0 or 5) and for 25 (if the number formed by the last two digits is 00, 25, 50, or 75). Questions involve applying these rules to quickly check for divisibility.

• Magic Number 15

  Yossi writes the number `15` on the board. Then, Danny adds a digit to the right and a digit to the left of the number written on the board, such that the new number is still divisible by `15`.

  Find this number. Is there only one possibility?

  Note: The digit added to the left is not zero.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Proof and Example -> Constructing an Example / Counterexample Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25 Number Theory -> Division
  Sources:
  • problems.ru web site

• Question

  How many numbers between `1` and `100` are not divisible by `2` or `5`?

  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 5 and 25 Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Set Theory Number Theory -> Division

• Question

  a. Prove that among `11` natural numbers, it is always possible to choose two such that their units digit is the same.

  b. Prove that among `11` natural numbers, it is always possible to choose two such that their difference is divisible by `10`.

  Topics:

  Combinatorics -> Pigeonhole Principle Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25

• Palindromic Number

  Find a four-digit palindromic number that is divisible by 25 and not divisible by 3.

  Note: A palindromic number is a number that does not change if its digits are read in reverse order. For example, the number 5775 is a palindromic number, and the number 5778 is not a palindromic number.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25 Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 5