Number systems: problems with an unknown base

Murad sees 13 + 15 = 31 on the board and laughs: “That’s wrong!” Leyla says: “Not so fast — you don’t know the base.” Indeed, if the numbers are written in base x, the equality becomes (x + 3) + (x + 5) = 3x + 1, so x = 7. Check: 13₇ = 10, 15₇ = 12, 31₇ = 22, and 10 + 12 = 22 ✓. In the four DİM entrance papers of 2025–2026, 4 of the 8 number-system tasks were such “unknown base” problems — both closed questions with 5 options and coded questions where you write the answer yourself. They all rest on one idea: the expanded form of a number (the lesson “Number systems: positional systems and binary”).

From the expanded form to an equation

1. 1
   Condition

   Find the largest digit in the equality: x must be bigger than it.

2. 2
   Expanded form

   Write every number with powers of x: from right to left 1, x, x², x³, …

3. 3
   Equation

   Collect like terms and solve: a linear equation directly, a quadratic or cubic one by factorising.

4. 4
   Selection

   Drop negative, zero and fractional roots, and roots not bigger than the largest digit.

5. 5
   Check and answer the question

   Convert the numbers to decimal with the x you found and check the equality. Then find exactly what is asked: x itself, the largest three-digit number, its decimal value and so on.

Finding the base from the last digit or the number of digits

When you convert a number to base b, the first remainder is its last digit. So if N written in base b ends in r, then N − r is divisible by b, and also b > r, because r is a digit. If two such conditions are given, the base is a common divisor of both differences; pick the common divisor that satisfies the digit condition.

The number of digits also tells you about the base. If N is written with exactly k digits in base b, it is not less than the smallest k-digit number (bᵏ⁻¹) and it is less than the smallest (k + 1)-digit number (bᵏ).

Consecutive digits, custom digit sets and the largest number

In base n the digits are 0, 1, …, n − 1. So if a task says “m and n are consecutive numbers”, then m = n − 1, the largest digit of the system. Digits can also be given as letters: the set {0, 1, 2, p, q, r, s} has 7 digits, so the base is 7, and the letters take the values p = 3, q = 4, r = 5, s = 6 in order. In both cases the largest k-digit number has every digit equal to the largest digit.

A program can solve such problems too: int(s, x) reads the string s as a base-x number (2 ≤ x ≤ 36), and the rest is checking the bases in a loop. The program below first finds the bases in which 29 ends in 5, then the base of 352ₓ + 544ₓ = 1006ₓ and the decimal value of the largest three-digit number in that base:

print([b for b in range(6, 30) if 29 % b == 5])
for x in range(7, 37):
    if int('352', x) + int('544', x) == int('1006', x):
        print(x, x**3 - 1)

[6, 8, 12, 24]
9 728

for x in range(6, 17):
    # int('154', x) reads '154' as a base-x number
    pass

7

0s and 1s in binary: the harder cases

In “Number systems: positional systems and binary” you learned to split a number into different powers of two: each power gives one 1, and the largest power fixes the length. In DİM tasks the number is often given as a difference, a square or in another base. Two rules let you open it up without long calculation.

This lesson completes the number-systems part. A short reminder: if the base is known, calculate with the methods of “Arithmetic in different number systems”; if it is unknown, build an equation from the expanded form and check the digit condition.