# Playground is giving random characters as answer

When I use playground to solve a math question, it outputs irrevelant characters. like this part of the answer.

THIS IS MY QUESTION:

Given that ‘a’ is a single digit, the remainder of dividing (25!) by (23 - a) is 60, what is ‘a’?

THIS IS PLAYGROUND ANSWER (check the end there is even hebrew characters)

Given a single-digit integer (a), the problem states that the remainder of dividing (25!) by (23 - a) is (60). We are to determine the value of (a).

First, we introduce (n = 23 - a). The condition says the remainder of (25!) when divided by (n) is 60. Therefore, we write the congruence:
[
25! \equiv 60 \pmod{n}
]

To find (a), we need to identify (n) such that this congruence holds true. Since (25!) is the factorial of 25, it is a large number with many factorizations.

Let’s check each possible value of (n), which range from 23 down to (14) (since (a) is a single digit between 0 and 9).

1. (n = 23)
[
25! \bmod 23
]
Since (25!) contains (23) as a prime factor, (25! \equiv 0 \pmod{23}). Therefore:
[
25! \not\equiv 60 \pmod{23}
]

2. (n = 22)
[
25! \bmod 22 = 0
]
because (22 = 2 \times 11) and both (2) and (11) are factors of (25!). Therefore:
[
25! \not\equiv 60 \pmod{22}
]

3. (n = 21)
[
25! \bmod 21 = 0
]
because (21 = 3 \times 7) and both (3) and (7) are factors of (25!). Therefore:
[
25! \not\equiv 60 \pmod{21}
]

4. (n = 20)
[
25! \bmod 20 = 0
]
because (20 = 2^2 \times 5) and (2) and (5) are factors of (25!). Therefore:
[
25! \not\equiv 60 \pmod{20}
]

5. (n = 19)
[
25! \bmod 19
]
Since 19 is a prime number:
[
25! \equiv -1 \pmod{19}
]
using Wilson’s theorem, because (25! = (19k + remainder)),
[
19k \equiv -1 \pmod{19} \Rightarrow remainder = 60 \pmod{19} \not \equiv 60.
]

6. (n = 18)
[
25! \bmod 18
]
because (18 = 2 \times 3^2) and both (2) and (3) are factors in (25!). So any factorial (>18) uniquely factors contains them and
[
\not\equiv 60 \pmod{18}
]

7. (n = 17)
[
25! \bmod 17 = -1
using (Wilson’s theorem \not 60
] here

8. (n = 16)
25! \bmod 16
we know (both)
so some calculation rules out

9. **(n=15)

10. **n=14)
nor solution.

Thus, only those checked multiple times each case uniquely only approach before.

Analyzing these in reachable unless checks out long all, hence zero suitable.

Thus final answer isolated is perhaps erroneously zero fault.

Hence final definitive expected analysis approach unless towards ( n=17 שהוא ממצא (only a). Thus ** Error uniquely traced check
Query since "deciding optimal

Hence final proposed towards intact ( n=15)=completed (a=)=Unique clauses( = 4.