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Exponents (Last Digit)

Explore deep-thinking maths problems involving last digits, powers, prime numbers, and digit tricks. Perfect for ages 11–16 and ideal for UKMT, AMC, and GCSE enrichment. Includes Olympiad-style theory with modular arithmetic, Diophantine equations, and factorial analysis.

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

Problems

  1. Find the last digit of the number:
    a) 2100
    b) 54949
    c) 20252025
  2. The Guinness Book of Records states that the largest known prime number is 23021337 − 1. Could this be a typo?
  3. A shop received 206 litres of milk in containers of 10 and 17 litres. How many containers of each type were there?
  4. Is the number 4730 + 3950 divisible by 10?
  5. Find the last digit in the product of all odd numbers from 1 to 2025.
  6. How many zeros are at the end of 2025 factorial?
    (2025! = 1 × 2 × 3 × ... × 2023 × 2024 × 2025)
  7. Prove that among the squares of any five natural numbers, you can always find two whose sum or difference is divisible by 10.
  8. Find the last digit of the number 77^7.
    (The powers are calculated from the top down: 77^7 = 7^(7^7))
  9. A number made of several sevens was written on the board: 777...77. Vlad erased the last digit of this number, multiplied the remaining part by 3, and added the erased digit back. He repeated this operation several times. Prove that eventually he will get the number 7.

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