What's the least positive integer that is not a factor of 25! and is not a prime number?

Question

It's more of a question of what's the definition of factor. To my understanding, for a number 25, 1,5,25 would be the factor because 1*25=25and 5*5=25. It is a question from GRE official guide book and the answer is 58. I don't have difficulty in understanding what the answer is doing, it says 58 is the least number that is not a prime and doesn't contain factors from 1 to 25. What makes me feel uncomfortable is how this answer is related to the question. Obviously the factor of 25! will be 1,2,...,25. And the least that is not a prime would be 4 because I remember we don't count 1 as a prime. Even if the answer is 58, shouldn't the question be framed as something like:

What is the least integer that is not a prime and has no factors same to the factor of 25!?

In fact, I put 26 as the answer because I think 25! has factor 1,2,...,25 but 26 is not one of these factors and it is also a prime. I think I misunderstood what the question is asking about but I can't figure out what exactly is the problem.

Answer 1

Too long for a comment, but answering some of your side questions. For the answer to the actual question, see the other answers already posted.

Being "not a factor of" and "having no factors the same" are two very different statements.

For positive integers $a,b$ the following statements are equivalent:

• $a$ is a factor of $b$
• $a$ divides $b$
• $b$ is a multiple of $a$
• $\dfrac{b}{a}$ is an integer
• There exists some integer $k$ such that $b = ka$
• $b\pmod{a}=0$
• $\vdots$

On the other hand "Shares a factor with" is something completely different. $a$ shares a factor with $b$ means that $\gcd(a,b)>1$.

It is also worth reminding you that $25!=1\cdot 2\cdot 3\cdot 4\cdots 23\cdot 24\cdot 25$ and that $25!$ has many factors greater than $25$., for example $2250$ is a factor of $25!$ since $2250 = 9\cdot 10\cdot 25$ and $(9\cdot 10\cdot 25)\cdot (1\cdot 2\cdots 7\cdot 8\cdot 11\cdot 12\cdots 23\cdot 24) = 25!$

Answer 2

Certainly $25!$ is divisible by all of $1,\ldots,25$. If $n$ is between $26$ and $50$ and not prime, it can be written as $ab$ where $a<b\le 25$ (except for $n=49$) and so divides $25!$. The same is true for $51,52,54,55,56$ and $57$. And $49\mid (7\times 14)\mid 25!$. So the only numbers from $26$ to $57$ inclusive not dividing $25!$ are primes. Of course $58$ has the prime factor $29$ and so cannot divide $25!$.

Answer 3

The least prime that is not a factor of $25!$ is $29$. Since we want it to be a composite, so the least composite number that is not a factor of $25!$ is $2 \times 29=58$.