Big Numbers Make Me Cry

Number Theory Level 3

What is the remainder when $2028!$ is divided by $2029$ ?

Details and assumptions

The factorial of a positive integer $n$ , denoted by $n!$ , is the product of all positive inegers less than or equal to $n$ . For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ .

The answer is 2028.

2 solutions

Brian Charlesworth
Oct 23, 2014

This is a direct application of Wilson's Theorem .

Since $2029$ is prime, we have that

$2028! \equiv -1 \mod{2029} \equiv 2028\mod{2029}$ .

Thus the remainder is $\boxed{2028}$ .

This is true for any case where $x!$ is divided by $x+1$ , where $x+1$ is prime. The remainder will be $x$ .

Joshua Ong - 6 years, 7 months ago

is -1 also a correct answer.

shivamani patil - 6 years, 7 months ago

Good point. Technically, yes it is, but in this type of question it is normally assumed that the desired answer is positive. Perhaps the phrasing of the question should have been more specific. At least you still got credit for answering the question; thankfully we get 3 guesses at these problems, unlike in an exam. :)

Brian Charlesworth - 6 years, 7 months ago

-1 is also correct

Monarch Adlakha - 6 years, 7 months ago

Victor Loh
Oct 23, 2014

Note that $2029$ is prime. By Wilson's Theorem,

$2028! \equiv -1 \pmod{2029} \equiv \boxed{2028} \pmod{2029}._\square$

Big Numbers Make Me Cry

The answer is 2028.

2 solutions

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