Lesson: Divisibility Rules

Comment on Divisibility Rules

Hi, are there any similar rules for 7? How do we determine if a large number is divisible by 7 and also by other prime numbers?
gmat-admin's picture

There are other rules for other numbers (like 7, 8, 11, etc), but I've never seen an official GMAT question that requires those rules. So, we have left them out.

Sure, thanks!
And thank you for this really wonderful video course!
gmat-admin's picture

Glad you like it!

Hey, how come we say that 0 is divisible by 4? I know that it has no remainder, but it doesn't make much sense to me. Say we have 8 cookies to be divided on 4 people, then each person gets 2 cookies and we have no remainder, so 8 is divisible by 4. But if we have 0 cookies and we want to divide that on 4 people, well, they'll be disapointed since they won't be getting any cookies, so we don't have anything to give, and therefore we don't have any remainder either. Any additional thoughts or any logical explanation? :-)
gmat-admin's picture

Hi Byefox,

20 is divisible by 4, because we can take 20 cookies and divide them into 4 EQUAL groups of 5 cookies each.

Likewise, 0 is divisible by 4, because we can take 0 cookies and divide them into 4 EQUAL groups of 0 cookies each.

Quote from the Official Guide for GMAT Review:

"If x and y are integers and x ≠ 0, then x is a divisor (factor) of y provided that y = xn for some integer n. In this case, y is also said to be divisible by x or to be a multiple of x.

For example, 7 is a divisor or factor of 28 since 28 = (7)(4), but 8 is not a divisor of 28 since there is no integer n such that 28 = 8n"

So, 12 is divisible by 4, since we can write 12 = (4)(3), where 3 is an integer.

Likewise, 44 is divisible by 4, since we can write 44 = (4)(11), where 11 is an integer.

AND, 0 is divisible by 4, since we can write 0 = (4)(0), where 0 is an integer.

Does that help?

Cheers,
Brent

I believe you have an error in this post - you eliminate A and say C is the correct answer, but at the end of the post you list A.

https://www.beatthegmat.com/on-a-weekend-6-college-friends-went-skiing-and-t300108.html
gmat-admin's picture

Good catch - I've edited my response accordingly.

Just to be clear, when we are looking at multiples in the GMAT, 0 and 1 are not a multiple of all numbers?
gmat-admin's picture

0 IS a multiple of all integers.

We say N is a multiple of integer d if there exists an INTEGER k so that N = kd

For example, 30 is a multiple of 5 because we can write 30 = 6(5)
Likewise, 21 is a multiple of 7 because we can write 21 = 3(7)
Likewise, 0 is a multiple of 7 because we can write 0 = 0(7)
Likewise, 0 is a multiple of 12 because we can write 0 = 0(12)
Likewise, 0 is a multiple of 55 because we can write 0 = 0(55)

1, on the other hand, is a multiple of 1 and -1 only

That said, when it comes to Integer Property questions, the GMAT typically restricts all values to POSITIVE integers.

Cheers,
Brent

https://gmatclub.com/forum/is-the-product-pqr-divisible-by-216152.html

So if I missed out on the tiny detail that pqr were not clearly said to be integers in this question and I applied to fact that 0 is a multiple of all integers to prove that both statements in this question were not sufficient, it would still have been a valid reasoning. Right?
gmat-admin's picture

Question link: https://gmatclub.com/forum/is-the-product-pqr-divisible-by-216152.html

Can you show me how you'd show that both statements in this question are not sufficient?

Cheers,
Brent

Hi Brent, referring to the divisibility rule for 3. can we say that the sum of all digits should be divisible by 3 with a condition that sum should not be 0. Since 0 is divisible by any number.


gmat-admin's picture

Good question!

We don't need a proviso that says the sum can't be zero. Here's why:

If the sum of the digits of integer N is zero, then we know that N = 0
Since 0 is divisible by 3, the rule still holds.

Having said all of that, when it comes to Integer Properties questions, the GMAT typically restricts the values to POSITIVE integers. So, the idea that 0 is divisible by 3 (or any other non-zero integer) does not come into play.

Does that help?

Cheers,
Brent

Question link :https://gmatclub.com/forum/number-xyz-is-the-product-of-positive-integer-n-and-9-is-x-y-z-204970.html

Thanks! so that means in the above Q'n I don't need statement 2 to get 0 out of the equation. My approach was to have x+y+z=0 as one possible value and then use statement 2 to get 0 out of the equation. But in GMAT 0 is out of the equation by default correct?
gmat-admin's picture

Question link: https://gmatclub.com/forum/number-xyz-is-the-product-of-positive-integer...

Be careful. We don't need to remove 0 as a possible value of XYZ, since the given information already does this for us.

We're told that "number XYZ is the product of POSITIVE INTEGER n and 9"
In other words, XYZ = 9n, where n is a POSITIVE INTEGER
So, XYZ cannot equal 0 (9 times a positive integer cannot equal 0)
In other words, x + y + z cannot equal 0

Does that help?

Cheers,
Brent

But what if the question said "n is any integer"? that includes 0 as well. So, in that case, do we need statement 2 to get rid of 0 as a possible case?
gmat-admin's picture

If the question read 'n is any integer), then we would need statement 2 (and statement 1) to get rid of 0 as a possible case.
That said, the GMAT won't require to know that zero is divisible by all non-zero integers.

Cheers,
Brent

Hi Brent, any rule for divisibility rule for 8? you mentioned it won't be required but I encountered one in the practice Q's
https://gmatclub.com/forum/10-25-560-is-divisible-by-all-of-the-following-except-126300.html

Similar to rule for 4, can we say if the last 2 digits are divisible by 8 then the number is divisible by 8?
gmat-admin's picture

Question link: https://gmatclub.com/forum/10-25-560-is-divisible-by-all-of-the-followin...

Another great question.
I've never seen an OFFICIAL question require us to know the divisibility rule for 8 (or for 7 or for 11).

That said, I've seen plenty of test prep companies create questions that hinge on the divisibility rule for 8 (and for 7 and for 11). The linked question is such a question.

The rule for divisibility by 8 is as follows:
If the number created by last 3 digits of integer N is divisible by 8, then N is divisible by 8.

Take, for example, the number 76,880
The number created by last 3 digits is 880
Since 880 is divisible by 8, we know that 76,880 is divisible by 8.

Likewise, we can take the number 1,233,128
The number created by last 3 digits is 128
Since 128 is divisible by 8, we know that 1,233,128 is divisible by 8.

Cheers,
Brent

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