Math Core

Base problems on the AMC 10 and 12 go well past converting numbers. You'll solve for an unknown base, test divisibility in another base, convert repeating "decimals" in base $b$ , count digits of enormous numbers, and untangle numbers whose digits are related in two different bases. All of it rests on one idea: a string of digits is a polynomial in the base.

Digits are a polynomial

$\underline{d_k}\,\underline{d_{k-1}} \cdots \underline{d_1}\,\underline{d_0}$ in base $b$ means

$d_k b^k + d_{k-1} b^{k-1} + \dots + d_1 b + d_0, \qquad 0 \le d_i \le b - 1.$

When the base is unknown, this turns a digit equation into an ordinary polynomial equation in $b$ . Don't forget the hidden condition: every digit must be less than $b$ .

Divisibility in base $b$

Since $b \equiv 1 \pmod{b - 1}$ , every power of $b$ is $\equiv 1$ , so a number is congruent to its digit sum modulo $b - 1$ . Since $b \equiv -1 \pmod{b+1}$ , powers of $b$ alternate $1, -1, 1, \dots$ , so a number is congruent to its alternating digit sum (starting from the units digit) modulo $b + 1$ . These are the base- $b$ versions of the tests for $9$ and $11$ .

Base-b facts

$\underline{d_k} \cdots \underline{d_0}$ in base $b$ equals $d_k b^k + \dots + d_1 b + d_0$ , with each $d_i \le b - 1$ .
Modulo $b - 1$ , a number is congruent to its digit sum. Modulo $b + 1$ , it is congruent to its alternating digit sum.
A positive integer $N$ has $k$ digits in base $b$ exactly when $b^{k-1} \le N \lt b^k$ .
A repeating block $0.\overline{d_1 d_2 \cdots d_k}$ in base $b$ equals $\dfrac{\underline{d_1} \cdots \underline{d_k}\ (\text{base } b)}{b^k - 1}$ .

The last fact works just like $0.\overline{36} = \dfrac{36}{99}$ in base ten: multiplying by $b^k$ shifts one full block left, and subtracting the original cancels the tail.

Counting digits of huge numbers

To count base-ten digits of something like $4^{50} \cdot 5^{96}$ , pair up $2$ s and $5$ s into $10$ s: $2^{100} \cdot 5^{96} = 2^4 \cdot 10^{96} = 16 \cdot 10^{96}$ , which has $2 + 96 = 98$ digits. For powers of $2$ in base $8$ or $16$ , group the exponent: $2^{100} = 2 \cdot 8^{33}$ , which lies between $8^{33}$ and $8^{34}$ .

Worked example: Solving for the base

In some base $b$ , $3 \times 25_b = 111_b$ . Find $b$ .

$3(2b + 5) = b^2 + b + 1$ , so $6b + 15 = b^2 + b + 1$ and $b^2 - 5b - 14 = 0$ , which factors as $(b - 7)(b + 2) = 0$ . So $b = 7$ . The largest digit used is $5$ , and $5 \lt 7$ , so this is valid. Check: $25_7 = 19$ , $3 \cdot 19 = 57 = 49 + 7 + 1 = 111_7$ .

Worked example: A remainder in base 8

What is the remainder when $3571_8$ is divided by $7$ ?

In base $8$ , a number is congruent to its digit sum modulo $7$ : $3 + 5 + 7 + 1 = 16 \equiv 2 \pmod 7$ . The remainder is $2$ . (Check: $3571_8 = 1913 = 7 \cdot 273 + 2$ .)

Worked example: A repeating fraction in base 7

Write $0.\overline{36}_7$ as a fraction in lowest terms.

The block $36_7 = 3 \cdot 7 + 6 = 27$ repeats with length $2$ , so the value is $\dfrac{27}{7^2 - 1} = \dfrac{27}{48} = \dfrac{9}{16}$ .

Worked example: Digits of a power

How many digits does $2^{100}$ have when written in base $8$ ?

$2^{100} = 2 \cdot 2^{99} = 2 \cdot 8^{33}$ . So $8^{33} \le 2^{100} \lt 8^{34}$ , and $2^{100}$ has $34$ digits in base $8$ : a $2$ followed by $33$ zeros.

Common mistake

After solving for a base, check the digits. The equation $23_b \times 2 = 51_b$ gives $4b + 6 = 5b + 1$ , so $b = 5$ algebraically. But a base- $5$ number can't contain the digit $5$ , so no base works. Also, $b$ must be an integer greater than $1$ ; discard negative or fractional roots.

Practice

Practice 1

Write $2026$ in base $7$ . (Enter the digits as a number.)

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

How many digits does $4^{50} \cdot 5^{96}$ have when written in base ten?

Practice 3

In some base $b$ , $46_b \times 2 = 103_b$ . Find $b$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

What is the remainder when the base-six number $543210_6$ is divided by $7$ ?

Practice 5

When $\dfrac{1}{5}$ is written in base $3$ , it is a repeating fraction. What is it?

Practice 6

How many positive integers less than $729$ are palindromes when written in base $3$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

A positive integer $N$ has three digits $\underline{a}\,\underline{b}\,\underline{c}$ in base $6$ , and the same digits in reverse order, $\underline{c}\,\underline{b}\,\underline{a}$ , in base $11$ . Find $N$ (in base ten).

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

What is the sum of the digits of $10^{40} - 2026$ (written in base ten)?

Real contest practice

2019 AMC 10A, Problem 18: find the base from a repeating base- $k$ fraction.
2025 AMC 12B, Problem 4: a number with related two-digit forms in base seven and base nine.
2013 AMC 12B Problems: look for the problem where Bernardo writes a number in base $10$ and base $15$ .

Digits are a polynomial

Divisibility in base bbb

Counting digits of huge numbers

Practice

Real contest practice

Divisibility in base $b$