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21 Oct 2012, 09:03
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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.
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.
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21 Oct 2012, 09:31
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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.
21 Oct 2012, 20:53
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danzig
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.
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
