10.4. Let x be some natural number. Among the statements:

$2 \times$ greater than 70

x less than 100

$3 x$ greater than 25

x not less than 10

x greater than 5

three are true and two are false. What is x
Answer: 9.

Solution

The first statement is equivalent to x > 35, and the third statement is equivalent to $\mathrm{x}>8$, since x is a natural number. The first statement is false, because if it were true, then the third, fourth, and fifth statements would also be true, i.e., there would be at least 4 true statements. Therefore, x does not exceed 35. From this, it follows that the second statement is true. But then the fourth statement is false (otherwise, the third and fifth statements would also be true, and again we would have at least 4 true statements). Therefore, statements 3 and 5 are true. And from the falsity of the fourth statement and the truth of the third statement, it immediately follows that $x=9$.