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Patterns in Numbers and Digits

Problems on periodicity. Spot patterns in them!

Subject: MathematicsCourse: Olympiad MathematicsAges: Junior, IntermediatePrimary age: Junior

Theory

Problems on periodicity.

Spot patterns!

Problems

A. Decimal expansions and repeating fractions

  1. Verify these decimal expansions:
    1. 1/3 = 0.(3)
    2. 1/6 = 0.1(6)
    3. 7/30 = 0.2(3)
    4. 7/11 = 0.(63)
  2. Find the 100th digit after the decimal point in the expansion of 1/7.
  3. Division by strings of 9s; purely periodic fractions:
    1. Compute by long division: 1 ÷ 9; 1 ÷ 99; 1 ÷ 9,999.
    2. Prove the general rule: 1/(99…9) = 0,(00…01), where the denominator has n nines and the repeating block has length n with (n−1) zeros followed by 1.
    3. Show that any purely periodic proper fraction equals a fraction whose numerator is the period and whose denominator is 10r − 1 (a number of r nines), where r is the period length.
  4. Convert to decimal form:
    1. 23/99
    2. 1234/999,999

B. Repunits and divisibility

  1. In the sequence 1, 11, 111, 1111, …, how many of the first 100 terms are divisible by 13?
  2. If a repunit 11…11 (all digits 1) is divisible by 7, prove that it is also divisible by 11, 13, and 15,873.
  3. In the sequence 1, 11, 111, 1111, …:
    1. Prove there exist two terms whose difference is divisible by 196,673.
    2. Deduce that there exists a repunit divisible by 196,673.
    3. For any natural number a not divisible by 2 or 5, prove there exists a natural b such that the product ab is a repunit (written using only the digit 1).
  4. If natural numbers a and m are coprime, prove there exists n such that an − 1 is divisible by m.

C. Cyclic digit moves and special base‑10 constructions

  1. A k-digit multiple of 13 has its first digit moved to the end. For which k is the resulting number still a multiple of 13?
    Example: 503,906 → 39,065 remains divisible by 13; 7,969 → 9,697 does not.
  2. Find a six‑digit decimal number that becomes 5 times smaller when its first digit is moved to the end of the number.
  3. The last digit of a (decimal) number is 2. If this digit is moved to the front, the number doubles. Find the smallest such number.
  4. A five‑digit number is divisible by 41. Prove that any cyclic permutation of its digits is also divisible by 41.
    Example: knowing that 93,767 is divisible by 41, conclude that 37,679 is divisible by 41.

D. Base‑16 (hexadecimal) curiosities

  1. Does there exist a hexadecimal number that, when multiplied by 2, 3, 4, 5, and 6, yields numbers that are permutations of its digits (in base 16)?
  2. Find a hexadecimal number that increases by an integer factor when its last digit is moved to the front (both operations understood in base 16).

E. Last digits and modular arithmetic

  1. Determine the last digit of 3377 + 7733.
  2. Find the last two digits of 22000.
  3. What is the last digit of 9,999,999,999?
  4. Find the last digit of 9999(99999999).

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