4. An ant walks on the numeric keypad of a mobile phone, which consists of 10 buttons arranged as shown in the figure. The ant always moves from the button it is on to an adjacent button horizontally or vertically; it starts from the button 1 and walks around the keypad for a while, finally stopping on the button 0, which it had never visited before. Consider the number $n$ obtained by concatenating the digits of the buttons the ant has visited, in the order in which it visited them (for example, the path in the figure would correspond to the number 12580). State how many of the following four statements are certainly true:

![](https://cdn.mathpix.com/cropped/2024_04_17_8e3f264af5d5452f36c1g-03.jpg?height=331&width=288&top_left_y=977&top_left_x=1592)

- " $n$ is not a multiple of 3 ";
- " $n$ is not a multiple of $8 $";
- " $n$ consists of an odd number of digits";
- " $n$ is not a square".
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
4. The answer is (D). The first statement is not necessarily true, since the ant can compose the number 14580, which is a multiple of 3 (the sum of its digits is 18). It is, however, true that $n$ cannot be a multiple of 8: we know that $n$ ends in 580, 780, or 980; if $n$ were divisible by 8, then $n-80$ would also be divisible by 8, but $n-80=100 \cdot d$, where $d$ is odd, so it is a multiple of 4 but not of 8. It is also true that $n$ consists of an odd number of digits: the ant always moves from an odd digit to an even digit and vice versa; the path starts on the key 1 (odd) and ends on the key 0 (even), so the ant must make an even number of moves, which corresponds to an odd number of keys pressed. Finally, it is also true that $n$ cannot be a square: since $n$ ends in 0, it is divisible by 10; but if $n$ were a square, it would then be divisible by 100 (and it is not, since it does not end with two zeros). In conclusion, three of the four statements are certainly true.