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

Hi Brent,

The following question is from OG-20:

If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following?

A: 3
B: 6
C: 7
D: 8
E: 10

Since it is given that y is a multiple of 5 and 3x + 4y = 200, this means that 5 and 3x + 4y = 200 are divisors of y. So if 5 can divide y this means that the units digits of y must be either 0 or 5. So from this, I concluded that x must be 10 since it is the only other number that can divide y assuming that the units digit of y must be zero.

The solution provided in the official guide is different from the way I solved it so I want to know whether my approach is valid?
gmat-admin's picture

I'm not sure about that approach. I have two questions about your solution:

I agree with your statement "if 5 can divide y this means that the units digits of y must be either 0 or 5", but I'm not sure how you then concluded that "x must be 10, since it is the only other number that can divide y assuming that the units digit of y must be zero."

1) How can we assume the units digit of y must be zero?
2) When you say "x must be 10." do you mean, "x must be a MULTIPLE of 10" or do you mean x = 10?

Cheers,
Brent

(1) I assumed that units digit of y must be zero by looking at the answer options. If we plug in the other answer options for x in 3x - 4y = 200, it won't be possible to solve the expression since it is set equal to 200. So I assumed any multiple of 10 could work.

(2) I meant to say 10 must be the answer among the options.
gmat-admin's picture

(1) "If we plug in the other answer options for x": I'd be careful about plugging in the answer choices, since the question asks "x must be a MULTIPLE of which of the following?" So, for example, for answer choice A, we can test x = 3, x = 6, x = 9, etc.

but we have 3x right? So if 3 cannot divide 200 so this means that any number times 3 cannot divide 200.

so testing multiples of answer options won't be conclusive I guess. only 10 makes sense to me.

I just thought of this now. please let me know if this line of thinking works?
gmat-admin's picture

If 3 cannot divide 200 so this means that any number times 3 cannot divide 200.

Yes, that's true.

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