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Re: If a and b are positive integers such that gcd(a,b)=13, then find the [#permalink]
ArunSharma12 wrote:
nick1816 wrote:
If a and b are positive integers such that gcd(a,b)=13, then find the sum of all possible distinct values of \(gcd(a^3,b)\).

gcd = Greatest common divisor

A. 13
B. 2197
C. 2210
D. 2379
E. Can't be determined


GCD(a,b)=13
\(a = 13^x * k \) (a has 13 one factor and some other integer not common to b)
\(b = 13^y * m\) (b has 13 one factor and some other integer not common to a)
such that minimum value between x and y has to 1.

if x = 1, the \(a^3 = 13^3 * k^3 \)
\(GCD(a^3,b)\):
b can take the following values
\(b_1 = 13*m\); \(GCD_1 = 13\)
\(b_2 = 13^2*m\); \(GCD_2 = 13^2\)
\(b_3 = 13^3*m\); \(GCD_3 = 13^3\)
after this if we increase the power of 13 in b, it'll start giving us the GCD as \(13^3\).

when x = 2, the \(a^3 = 13^6 * k^3 \)
\(GCD(a^3,b)\):
b can take the following values
\(b_1 = 13*m\); \(GCD_1 = 13\)

sum of all possible distinct values of gcd(a3,b) = \(13 +13^2 + 13^3\) = 2379
Ans: D


Hi, can you please explain that for case when x = 2, why i can't take the value \(b_4.\\
\\
For example,\\
\\
b_4 = 13^4*m\); \(GCD_4 = 13^4\)
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Re: If a and b are positive integers such that gcd(a,b)=13, then find the [#permalink]
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funkyakki wrote:
Hi, can you please explain that for case when x = 2, why i can't take the value \(b_4.\\
\\
For example,\\
\\
b_4 = 13^4*m\); \(GCD_4 = 13^4\)


Arun wrote:
GCD(a,b)=13
\(a=13^x∗k\) (a has 13 one factor and some other integer not common to b)
\(b=13^y∗m\) (b has 13 one factor and some other integer not common to a)
such that minimum value between x and y has to 1.

when x = 2, the \(a^3=13^6∗k^3\)
\(GCD(a^3,b)\)
b can take the following values
\(b_1=13∗m; GCD_1=13\)

Hi funkyakki,
here b can not take any higher powers.
lets take the case,
\(b_2=13^2∗m; GCD_2=13^2\)
minimum value of x,y is 2.
once we focus on GCD value, we can see that GCD is now 13^2 and it violates the constraint given in the question stem: gcd(a,b)=13.
Hence for any higher powers of x = 2,3,4... there will be only one value for y, y =1.
This will also limit the GCD value of \(GCD(a^3,b)\) to 13 thereby creating no more distinct GCD values.
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Re: If a and b are positive integers such that gcd(a,b)=13, then find the [#permalink]
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Basic Concept: if we raise any given integer to a positive integer Power, the UNIQUE Prime Factors that make up that number will not change ——-> only the amount of those Unique Prime Numbers will change

If the GCF of (A , B) = 13

Then when we raise A to the 3rd Power and keep B the same, the same Unique Prime Factors that divide into A and B will be there. All that will change is the amount.


Case 1:

A = 13x

B = 13y

Since GCF = 13, x and y are co-prime

When we raise A to the 3rd Power, the two values will still only share a Factor of 13


Case 2:
A = 13x
B = (13)^2 y

Where x and y are co-prime such that the GCD(A,B) = 13

Right now all A and B shares is the common factor of one 13

However when we Cube A, there will be 3 more prime factors of 13 that make up A

The GCF will now become (13)^2


Case 3:
A = 13x
B = (13)^3 y

Originally, like the others, the GCF of A and B = 13 ———- the only factor they share is a 13

However if we Cube A ———> (A)^3 will now have THREE 13s and the new Greatest Shared Factor will be (13)^3


Sum of all possible GCFs:

13 + (13)^2 + (13)^3 = 2,379

D

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Re: If a and b are positive integers such that gcd(a,b)=13, then find the [#permalink]
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Re: If a and b are positive integers such that gcd(a,b)=13, then find the [#permalink]
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