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Ravixxx
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If u=2^{5}-2, v=3^{5}-3 and w=4^{5}-4, what is the greatest common div 07 May 2020, 00:51
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25% (medium)

Question Stats:

77% (02:06) correct 23% (02:13) wrong based on 57 sessions

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If \(u=2^{5}-2\), \(v=3^{5}-3\) and \(w=4^{5}-4\), what is the greatest common divisor of u,v and w?

(A) 20
(B) 30
(C) 40
(D) 50
(E) 60
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B
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If u=2^{5}-2, v=3^{5}-3 and w=4^{5}-4, what is the greatest common div 07 May 2020, 03:07
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Ravixxx
If \(u=2^{5}-2\), \(v=3^{5}-3\) and \(w=4^{5}-4\), what is the greatest common divisor of u,v and w?

(A) 20
(B) 30
(C) 40
(D) 50
(E) 60
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\(u=2^{5}-2 = 32-2 = 30\)

i.e. answer can NOT be options C, D and E which we can eliminate now

Also, Option A can be eliminated because 20 is not the factor of u so it can NOT be HCF of u, v and w



Hence, Answer: Option B

However, with confusing options we need to do the following calculations

\(v=3^{5}-3= 3*(3^4-1) = 3*(3^2+1)*(3^2-1) = 3*10*8 \)

\(w=4^{5}-4= 4*(4^4-1) = 4*(4^2+1)*(4^2-1) = 4*17*15 \)

so GCD of u, v and w = 30
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If u=2^{5}-2, v=3^{5}-3 and w=4^{5}-4, what is the greatest common div 07 May 2020, 05:23
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Ravixxx
If \(u=2^{5}-2\), \(v=3^{5}-3\) and \(w=4^{5}-4\), what is the greatest common divisor of u,v and w?

(A) 20
(B) 30
(C) 40
(D) 50
(E) 60
Show more

we can individually simplify each term also.
u = \(2^5 - 2 = 2(2^4 - 1) = 2(2^2 + 1)(2^2 - 1) = 2*5*3 = 2^1 * 3^1 * 5^1\)
v = \(3^5 - 3 = 3(3^4 - 1) = 3(3^2 + 1)(3^2 - 1) = 3*10*8 = 2^4 * 3^1 * 5^1\)
w = \(4^5 - 4 = 4(4^4 - 1) = 4(4^2 + 1)(4^2 - 1) = 4*17*15 = 2^2 * 3^1 * 5^1 * 17^1\)

therefore, the GCD(u, v, w) = \(2^1 * 3^1 * 5^1 = 30\) (option B)
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