shasadou
For nonnegative integers x, y, and m, what is the greatest value of m for which x^m is a factor of y!?
(1) y=x−1
(2) x is a prime number
Hi,
A Good Q..
the Q statement is
"For nonnegative integers x, y, and m, what is the greatest value of m for which x^m is a factor of y!?"What doe sthis mean..
It means the largest power of x that is there in y!..
formula is \(\frac{y}{x} + \frac{y}{x^2}..\) and so on till the fraction \(\frac{y}{x^z}\) becomes less than 1..
x is a prime number or the biggest prime in any integer..
lets see the sentences..
(1) y=x−1this means \(x^m\) in (x-1)!..
\(\frac{y}{x} + \frac{y}{x^2}..\). and so on means \(\frac{(x-1)!}{x}\)..
if x is prime, answer is 0..
if not it will depend on x..
say x=6, so y=5..
check for 3s in 5! as 3 is the largest prime number in 6..
\(\frac{5}{3}\)=1 so m=1..
different answers
But we do not know if x is prime or what is the largest prime in the integer x..
insuff
(2) x is a prime numbersince there is no corelation in y and x, we cannot answer ..
say y is 50 and prime is 5, then it is \(\frac{50}{5}+\frac{50}{25}=12.\).
and say 4 and prime is 3then 4/3=1..
insuff..
combined .
we know that x is prime and y is x-1..from this it becomes clear that y! or (x-1)! will not have x..so power of x will be 0, or m=0..
suff
C