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13 Apr 2020, 07:36
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D
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Difficulty:
95%
(hard)
Question Stats:
23% (02:27) correct
77%
(01:37)
wrong
based on 128
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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 = Greatest common divisor
A. 13
B. 2197
C. 2210
D. 2379
E. Can't be determined
ArunSharma12
Joined: 25 Oct 2015
Last visit: 20 Jul 2022
Posts: 512
Given Kudos: 74
Location: India
Schools: IIML-IPMX '22 (A)
GMAT 1: 650 Q48 V31
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13 Apr 2020, 09:04
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nick1816
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
18 Apr 2020, 19:19
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a = 13^m.x
b = 13^n.y
as GCD(a,b) is 13, either m or n will be 1.
GCD(a^3,b) = ?
i.e. GCD((13)^3m.x,13^n.y) = ?
when m=1, n can be 1,2,3
GCD(13^3.x, 13.y) = 13
GCD(13^2.x, 13^2.y) = 13^2
GCD(13^3.x, 13^3.y) = 13^3
when n=1, whatever power m takes, GCD(13^3m.x,13^n.y) will always be 13.
Hence there are 3 possible values 13, 13^2, 13^3.
13+13^2+13^3 = 2379
b = 13^n.y
as GCD(a,b) is 13, either m or n will be 1.
GCD(a^3,b) = ?
i.e. GCD((13)^3m.x,13^n.y) = ?
when m=1, n can be 1,2,3
GCD(13^3.x, 13.y) = 13
GCD(13^2.x, 13^2.y) = 13^2
GCD(13^3.x, 13^3.y) = 13^3
when n=1, whatever power m takes, GCD(13^3m.x,13^n.y) will always be 13.
Hence there are 3 possible values 13, 13^2, 13^3.
13+13^2+13^3 = 2379
