Question

Let S be the set of numbers of the form n(n + 1)(n + 2)(n + 3)(n + 4), where n is any positive integer. The first few terms of S are:
1*2*3*4*5 = 120
2*3*4*5*6= 720
3*4*5*6*7= 2520
and so no . What is the GCD of the elements of S?

Answer 1

Greatest common factor for s, i think it will be the numbers used for the 5 times table

Answer 2

emmm.. if I put this in form n(n + 1)(n + 2)(n + 3)(n + 4) = S and 5 times the table that is to say what should be the value of n?, it's easy but the neck of answering or interpretation ....am not able to understand

Answer 3

\(\large \color{black}{\begin{align}k=n(n+1)(n+2)(n+3)(n+4)\end{align}}\)


\(\large \color{black}{\begin{align} \begin{array}{|c|c|c|} \hline
\text{for n=1} & k=120 &k=2^3\times 3\times 5\\ \hline
\text{for n=2} & k=720&k=2^4\times 3^2\times 5\\ \hline
\text{for n=3} & k=2520&k= 2^3\times 3^2\times 5\times 7\\ \hline
\text{for} n=\cdots & k=\cdots&k=\cdots\\ \hline
\end{array}
\end{align}}\)
from the table

\(\large \color{black}{\begin{align}\gcd\left(n(n+1)(n+2)(n+3)(n+4)\right)=2^3\times 3\times 5=120 \end{align}}\)

Answer 4

product of any \(n\) consecutive integers is divisible by \(n!\)

Answer 5

120 is wrong ?

Answer 6

120 is right !

Answer 7

I'm a bit confused about GCD and GCF
What is the difference?

Answer 8

\(\large \color{black}{\begin{align}\gcd=\text{gcf}= \text{hcf}\end{align}}\)

Answer 9

http://en.wikipedia.org/wiki/Greatest_common_divisor

Answer 10

http://mathworld.wolfram.com/GreatestCommonDivisor.html

Answer 11

thanks - basically they are the same thing

Answer 12

factor = divisor

yes they are exact same thing

Answer 13

thanks you guys are real fast in solving
@ganishie8 and @ mathmath333

Answer 14

I was thinking of LCD where D means the denominator

Answer 15

\[b=ac\]

\(a\) is a `divisor` of \(b\)
\(a\) is a `factor` of \(b\)
\(b\) is a `multiple` of \(a\)

Answer 16

also notice \(0\) cannot be a divisor of any natural number because \(x= 0y\) is possible only when \(x=y=0\)

Answer 17

cool!