# Number theory, divisibility by 22

• Sep 12th 2006, 07:01 AM
suedenation
Number theory, divisibility by 22
Hi guys, I'm taking an advanced algebra course and this is one of the question in my assignment.

Q: Decide which of the following integers are divisble by 22.
a)0
b)192544
c)444
d)-32516
e)-195518

I don't know if I can just use the long division to do it or I have to apply the divisibility by 11. Thanks for your help. :)
• Sep 12th 2006, 07:07 AM
ThePerfectHacker
Quote:

Originally Posted by suedenation
Hi guys, I'm taking an advanced algebra course and this is one of the question in my assignment.

Q: Decide which of the following integers are divisble by 22.
a)0
b)192544
c)444
d)-32516
e)-195518

I don't know if I can just use the long division to do it or I have to apply the divisibility by 11. Thanks for your help. :)

Note,
gcd(2,11)=1
Thus, it is equivalent to saying that it is divisible by 2 and by 11.
Note all are even.
Thus, check those that are divisible by 11, how?
Check the alternate sum.
For example,
(b)
4-4+5-2+9-1=11
Which is divisible.
• Sep 12th 2006, 07:41 AM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
Note,
gcd(2,11)=1
Thus, it is equivalent to saying that it is divisible by 2 and by 11.
Note all are even.
Thus, check those that are divisible by 11, how?
Check the alternate sum.
For example,
(b)
4-4+5-2+9-1=11
Which is divisible.

Just dividing by 11 also is not rocket science.

RonL
• Sep 12th 2006, 08:23 AM
ThePerfectHacker
Quote:

Originally Posted by CaptainBlank
Just dividing by 11 also is not rocket science.

??:confused:
• Sep 12th 2006, 02:23 PM
Quick
Quote:

Originally Posted by ThePerfectHacker
??:confused:

he meant 22.

Would 0 be considered divisible? (I should know this but I'm too lazy to look it up elsewhere)
• Sep 12th 2006, 04:30 PM
ThePerfectHacker
Quote:

Originally Posted by Quick
he meant 22.

Would 0 be considered divisible? (I should know this but I'm too lazy to look it up elsewhere)

Yes!
The formal definition of divisibility is that,
a divides b means there exists a k such as,
b=ak
For the integers.

Thus, if we take k=0 then it answers they questions.