Lesson: Integer Properties Strategies

Comment on Integer Properties Strategies


Could you help with the Exercise #161 of OG17?

gmat-admin's picture

Glad to help!

Here's my step-by-step solution: https://gmatclub.com/forum/club-x-has-more-than-10-but-fewer-than-40-mem...



Thank you!


why is 51 not a prime number?
gmat-admin's picture

51 = 3 x 17
The sum of the digits = 5 + 1 = 6
Since 6 is divisible by 3, we know that 51 is divisible by 3


Hi! My question is pertaining to the rule regarding consecutive integers (N consecutive integers ==> 1 number must be divisible by N).... Does this mean that exactly 1 number within the list must be divisible by N, or at least one number (possibly more than 1?) must be divisible by N? I understood it as exactly 1 number must be divisible by N, but I just wanted to confirm.
gmat-admin's picture

Your interpretation is correct; in a set of N consecutive integers, exactly 1 of those integers is divisible by N.

Hi Brent,

The below question is from MGMAT, need your help in this one. Also if you could suggest more of such questions to practise.

If [x] denotes the least integer greater than or equal to x and [x/2]=0, which of the following could be the value of x ?

gmat-admin's picture

Came across this on a practice test:

What is the tens digit of the positive integer r?
(1) The tens digit of r/10 is 3
(2) The hundredth digit of r 10r is 6

What the best approach for this?
gmat-admin's picture


How can we pick random numbers to prove / disprove the statements here?
gmat-admin's picture

I have shown how we can test numbers in my solution here: https://gmatclub.com/forum/if-a-b-and-c-are-integers-and-abc-0-is-a-2-b-...


HI! Could you please help me solve the following question?

gmat-admin's picture

Here's my full solution: https://gmatclub.com/forum/m27-184477.html#p2289874


Hi Brent, could you help me with this question please? Thank you!! x

If a, b, k, and m are positive integers, is a^k a factor of b^m?
(1) a is a factor of b.
(2) k ≤ m.
gmat-admin's picture


y=13; is there any quick way to find y=13? Imho,It's a time consuming
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-x-and-y-are-integers-greater-than-3-and-15...

Once we know that 4(2 – y) is a multiple of 11, we can start testing possible scenarios.
Some possible multiples of 11 include 11, 22, 33,...etc as well as 0, -11, -22,... etc

Important: Since 4 is NOT a multiple of 11, it must be the case that (2 – y) is a multiple of 11.

Now let's test some possible scenarios...

Let's start with (2 - y) = 11
This case, y = -9
However, we're told that y is an integer greater than 3

What about (2 - y) = 22?
This case, y = -20, but we're told that y > 3
At this point, we can see that, if we keep testing bigger and bigger multiples of 11 (33, 44, 55 etc), the corresponding y value will always be negative.

So let's try some smaller multiples of 11

What about (2 - y) = 0?
This case, y = 2, but we're told that y > 3

What about (2 - y) = -11?
This case, y = 13
PERFECT! This satisfies the condition that y > 3

What about (2 - y) = -22?
This case, y = 24
This also satisfies the condition that y > 3

At this point we can see that 13 is the smallest possible y-value such that (2 – y) is a multiple of 11.

So, once we know that (2 – y) is a multiple of 11, we don't have to test many values.




Approach please.

Hi Brent,

Could you please help me solve these 3 questions?

1)In a certain deck of cards, each card has a positive integer written on it. In a multiplication game, a child draws a card and multiplies the integer on the card by the next larger integer. If each possible product is between 15 and 200, then the least and greatest integers on the cards could be?

2)In a certain board game, a stack of 48 cards, 8 of which represent shares of stock, are shuffled and then placed face down. If the first 2 cards selected do not represent shares of stock, what is the probability that the third card selected will represent a share of stock?

3) At a speed of 50 miles per hour, a certain car uses 1 gallon of gasoline every 30 miles. If the car starts with a full 12 gallon tank of gasoline and travels for 5 hours at 50 miles per hour, the amount of gasoline used would be what fraction of a full tank?

Thank you!!

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