## Number properties - is 0 a multiple of every integer?

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thinkblue

### Number properties - is 0 a multiple of every integer?

I always get confused about how to think of 0. Please tell me if the following are correct -

1. 0 is neither positive nor negative.
2. 0 is even.
3. 0 is a multiple for every integer.
4. 0 is not a prime.
5. Is 0 a factor of every number? I think not but just checking.

Also,

-> Can negative numbers be prime?
-> If an int has only 2 factors then we know it is prime right away, correct?
-> Special sums in Chap 4 (page 43) only applies to consecutive integers. But not consecutive evens or consecutive primes or consecutive odd correct?
-> Average and median of consecutive numbers are equal correct??

Finally!

If faced with a very large number, is there an easy way to tell it is a prime number?? Or a multiple of 7 or 13 or 17??
All this is assuming it is not divisible by 2,3,4,5,6,8,9,10 of course.
Guest

hi thinkblue,

yes, zero is a multiple of every number. the trick for the GMAT however is to read carefully what KIND of multiple they ask for. so while yes, zero is a multiple of every number, it is not a POSITIVE multiple of ANY number. usually on the GMAT they say "positive multiples of" or some other clarifying language so that they generally don't want you to include 0 (not guaranteeing that, but so far I haven't seen them wanting you to include zero as a multiple).

the other thing to be careful of is to remember that while zero is a multiple of every number, zero is NOT a FACTOR of any number except zero. since any number times zero is zero, there is no other number that could have a zero as a factor. to put it another way, for a number to NOT be zero, zero must NOT be one of its factors.

"Can negative numbers be prime?" Nope. Primes are always positive.
"If an int has only 2 factors then we know it is prime right away, correct?" If the integer is positive, then yes.
"Special sums in Chap 4 (page 43) only applies to consecutive integers. But not consecutive evens or consecutive primes or consecutive odd correct?" This I'm not sure of.
"Average and median of consecutive numbers are equal correct??" Yes.

"If faced with a very large number, is there an easy way to tell it is a prime number?? Or a multiple of 7 or 13 or 17?? All this is assuming it is not divisible by 2,3,4,5,6,8,9,10 of course."

I don't believe there's an easy way to figure out if a number is prime. My guess is for the GMAT you won't have to deal with number so large, but even then you'd only have time to maybe test and see if the number is divisible by 2, 3, 5, 7 or 9 and just guess.

Hope this helped. Hopefully an instructor can come along and check what I said, but I think we're on the right track here.
san

Now

0 = 0 * n

It means every number is factor of zero including zero.

Factor for zero are undefined.
esledge
ManhattanGMAT Staff

Posts: 897
Joined: Tue Mar 01, 2005 6:33 am
Location: St. Louis, MO

To the Guest who posted the detailed reply, thanks, that was great!

The only statement on which I disagree is:
the other thing to be careful of is to remember that while zero is a multiple of every number, zero is NOT a FACTOR of any number except zero.

(Emphasis with italics added by moderator)

Zero is not a factor of anything, period. Look at it this way: for y to be a factor of x, you would check whether x/y = integer. For example, 3 is a factor of 15 because 15/3 = 5, an integer. But n/0 = undefined, not an integer. Thus zero cannot be a factor of n, regardless of n's value. You cannot say that 0/0 = 0 just because zero is the numerator. The denominator of 0 makes 0/0 undefined.

I strongly agree with your point that:
usually on the GMAT they say "positive multiples of" or some other clarifying language so that they generally don't want you to include 0 (not guaranteeing that, but so far I haven't seen them wanting you to include zero as a multiple).

I have yet to run across a problem where the answer hinges on these properties of zero.

-> Special sums in Chap 4 (page 43) only applies to consecutive integers. But not consecutive evens or consecutive primes or consecutive odd correct?

The special sums rule applies to any evenly spaced set, which would include consecutive integer, consecutive evens, consecutive odds, consecutive multiples of 11, etc.

The reason: if you pair the highest and lowest values, then pair the 2nd-highest and 2nd-lowest values, etc. the sum of each pair is constant. In fact, the rule would even work on a symmetrically spaced set, even if the difference between terms did vary. For such an example, see 10th ed. OG PS#176.

You CANNOT use the special sum rule for consecutive primes, as the spacing of consecutive primes increases as the value of the primes increases. Consecutive primes are neither evenly spaced, nor symmetrically spaced.

If faced with a very large number, is there an easy way to tell it is a prime number?? Or a multiple of 7 or 13 or 17??

The easiest way to tell whether a large number is divisible by a "weird" prime like 7, 13, or 17 (i.e. you don't have an easy number property to check)? Long division, actually.

The GMAT will not ask you to determine whether a really large specific number is prime. However, you might have to notice some cue that indicates a large number is NOT prime, i.e. that x has some other factor between 1 and x. You might also have to know very generally that among very large numbers (e.g. all numbers > 10,000), there are both primes and non-primes.
Emily Sledge
Instructor
ManhattanGMAT