need help with another prove

[david90]

The product of four consecutive integers is div. by 24.

 

[eltonjohn]

i think that the trick is to express tha suite of consecutive numbers in powers of two

two cases

S1 = (p.2^n )*(p. 2^n+1 )*(p.2^(n+1) *(p.2^(n+1) +1)
and
S2 = (o.2^m +1 )*(o. 2^(m+1) ) * (o.2^(m+1 )+1) *(o.2^(m+2) )

p and o are any intergers

S1 a even suite (starts with a even number
S2 an Odd suite (starts with a odd number)

 

[klug]

It is very simple:

1. Every second consecutive integer is divided by 2.
2. Every third consecutive integer is divided by 3.
3. Every forth consecutive integer is divided by 4=2*2.

So, you see, there will be 1 integer divided by 4, at least 1 integer divided by 3, two integers divided by 2.

As 4 is divided by 2, then there will be 1 integer that will be divided by 2 and will not divided by 4.

So, there will be these known component dividers in result product: (2*2)*2*3=24, thus this product will be divided by 24.

Sorry, my mathematical language is poor - I never used English in mathematics.

 

[waterman]

Here is my proof.