+283 What is the largest integer n such that 3^n is a factor of 1*3*5*...*97*99?
Listfor(n, 1, 165, (99!! % 3^n) =(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 508 3731656658, 508 3731656658, 5083 7316566580, 11946 7693931463, 53124 9958120761, 114892 .......etc.
I see that the first 26 powers of 3 are factors of 99!!. Therefore, the largest integer n = 26, or:
3^26 =2,541,865,828,329 which is a factor of 99!!.
Note: 99!! is the double factorial function, i.e., 99 x 97 x 95, 93 x.............x 5 x 3 x 1.
+33661 99!→2^95*3^48*5^22*7^16*11^9*13^7*17^5*19^5*23^4*29^3*31^3*37^2*41^2*43^2*47^2*53*59*61*67*71*73*79*83*89*97
n is 48.
99! is 99*98*97*...*3*2*1 It is a single factorial, not a double factorial.
+33661 Oops! I read it as 1*2*3*...*97*98*99. It isn't!
With z, say, as the product of the odd integers from 1 to 99 we have:
z→3^26*5^12*7^8*11^5*13^4*17^3*19^3*23^2*29^2*31^2*37*41*43*47*53*59*61*67*71*73*79*83*89*97
so n is 26
(and, yes, this is known as a double factorial. Thanks for pointing out my error Guest.).
+283 Thanks for all of y'alls help, the answer is 26, I think mathexpertise got confused somewhere.
+26393 What is the largest integer n such that 3^n is a factor of 1*3*5*...*97*99?
\(\begin{array}{|rcll|} \hline 99!! &=& 1\cdot 3 \cdot 5\cdot 7\cdot \ldots \cdot 95\cdot 97\cdot 99 \\\\ &&\text{Formula:}~\boxed{(2n-1)!! = \frac{(2n)!}{2^n\cdot n!}} \\\\ && (2\cdot 50 - 1)!! = \dfrac{(2\cdot 50)!} {2^{50}\cdot 50! }\\\\ 99!! &=& \dfrac{100!} {2^{50}\cdot 50! } \\ \hline \end{array}\)
100!
The factor \(3^i\)
\(\begin{array}{|rcll|} \hline \frac{100}{3} &=& [33].\overline{3} \\ \frac{100}{3^2} &=& [11].\overline{1} \\ \frac{100}{3^3} &=& [3].\overline{703} \\ \frac{100}{3^4} &=& [1].\overline{234567901} \\ \frac{100}{3^5} &=& 0 \\ \hline 33+11+3+1 &=& 48 \\ i &=& 48 \\ \hline \end{array} \)
50!
The factor \(3^j\)
\(\begin{array}{|rcll|} \hline \frac{50}{3} &=& [16].\overline{6} \\ \frac{50}{3^2} &=& [5].\overline{5} \\ \frac{50}{3^3} &=& [1].\overline{851} \\ \frac{50}{3^4} &=& 0 \\ \hline 16+5+1 &=& 22 \\ j &=& 22 \\ \hline \end{array} \)
\(\begin{array}{|rcll|} \hline && \dfrac{3^{48}}{3^{22}} \\\\ &=& 3^{48-22} \\ &=& 3^{26} \\ \hline \end{array} \)
\(\text{The largest integer $n$ such that $3^n$ is a factor of $1*3*5*...*97*99$ is $\mathbf{26}$ }\)
