#include <stdio.h>
#include <stdlib.h>
 
int main() {
    char string[100];
    char tabela[256];
 
    // Limpa a tabela.
    for (int i = 0; i < 256; i++) {
        tabela[i] = 0;
    }
 
    // Lê a frase do usuário.
    printf ("\nEscreva um texto: \n");
    fgets(string, 100, stdin);
    int tamanho = strlen(string);
 
    // Monta a tabela de frequências.
    for (int i = 0; i < tamanho; i++) {
        char c = string[i];
 
        // Ignora o espaço.
        if (c == ' ') continue;
 
        // Se for uma letra minúscula, troca por maiúscula.
        if (c >= 'a' && c <= 'z') c = c - 'a' + 'A';
 
        // Contabiliza na tabela.
        tabela[c]++;
    }
 
    // Busca o índice de maior ocorrência na tabela.
    int maior = 0;
    char letra = '\0';
    for (int i = 0; i < 256; i++) {
        int t = tabela[i];
        if (t > maior) {
            maior = t;
            letra = (char) i;
        }
    }
 
    // Mostra o resultado.
    printf("\nO caractere '%c' aparece %d vezes.\n", letra, maior);
 
    // Fim.
    return 0;
}

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O rato roeu a roupa do rei de Roma e a rainha roeu o resto.