Divisibility question

Question

Please, check my method to analyze whether an integer is divisible by other.

Q: If x and y are positive integers, is x divisible by y?
Method:
I calculate the prime factorization of x and y
x: \(a^2.b.c\)
y: \(a^2.b\)
Being a, b, and c prime factors.
So, \(\frac{x}{y} = \frac{a^2.b.c}{a^2.b}\)
We can eliminate a^2.b in the dividend and the divisor.
So, c is the answer, and x is divisible by y.

Here, is my question:
When I cannot eliminate a prime factor of the divisor "y" (because there is not the same prime factor in the dividend "x"), in that case, x is not divisible by y, right?

Please, confirm whether my reasoning is Ok and why. Thank you very much.

SOURH7WK · Answer

Yes, you are right. In other words u can say y has to be a factor of x. If there are different prime numbers in numerator & denominator and u can not cancel out the demo, then it's not divisible. Ex - 2/3, 7/5, etc.

Or u can express it in this form x= p*y, where p is an integer.

Or u can express it in this form x= p*y, where p is an integer.

KarishmaB · Answer

Yes, perfect. Divisibility and factorization are closely related because factors are also called divisors!

What are the factors of 10? 1, 2, 5 and 10 
What are the divisors of 10? 1, 2, 5 and 10 - these numbers divide 10 completely or in other words, they are factors of 10.

People get confused with the term divisor because it is the number that divides another even when it is not completely divisible. But when you consider the divisor as 'divisor of a number', you only consider those numbers which completely divide.

When you say 'is a divisible by b?', you are asking whether b is a factor of a. So all you need to check is whether b is a factor of a. Prime factorization isn't required but if it is already given to you, good for you!

e.g.
a = 72, b = 24 
Is a divisible by b?
a = 72 = 24*3
Since 24 is a factor of a, a is divisible by b

or

a = 2^3*3^3 
 b = 2^3*3

You see that 2^3*3 = 24 is present in a so a is divisible by b.

On the other hand, if you have something like this:

a = 2^3*3^3 
 b = 2^3*5

Now, is b a factor of a? No because a has no 5. So a is not divisible by b.

Divisibility question

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards