Round numbers (2)

In this exercise, we say that a natural number is round in base $b$ , when the sum of of its digits in base $b$ equals its number of digits in this base.

For example, the number 34 is not round in base $10$ ( $3 + 4 \neq 2$ ), but it is round in base $3$ , as $1\cdot 3^3 + 0\cdot 3^2 + 2 \cdot 3^1 + 1 \cdot 3^0 = 34 \text{ and } 1 + 0 + 2 + 1 = 4.$ As another example, $511$ is not round in base $16$ as $1 \cdot 16^2 + 15 \cdot 16^1 + 15 \cdot 16^0 = 511 \text{ and } 1 + 15 + 15 = 31 \neq 3,$ but it is round in base $2$ (it has $9$ ones, that add up to $9$ ). Another example: $370273$ is not round in base $2$ , neither in base $3$ , …, however it is round in base $608$ , because $1 \cdot 608^2 + 1 \cdot 608^1 + 1 \cdot 608^0 = 370273 \text{ and } 1 + 1 + 1 = 3.$

A sequence of pairs of natural numbers $(n,b)$ , where $n$ is a natural number and $b\geq 2$ , is called bi-round if it does contain at least two pairs $(n,b)$ with the property that $n$ is round in base $b$ .

Write a program that, given a sequence of pairs of natural numbers, determines whether it is bi-round or not.

Your program must include, use and implement the function

    bool round (int n, int b);

that indicates if a natural number is round on base $b$ or not.

Input

The input is a non-empty sequence of pairs of natural numbers $(x,b)$ with $b\geq 2$ .

Output

The program has to write if the input sequence is bi-round or not.

Please follow the format described in the examples. Your code should follow the style rules and include the appropriate comments.

Problem information

Author: Unknown
Translator: Maria Serna

Generation: 2026-01-25T16:58:38.490Z