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Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
30^20 – 20^20 is divisible by all of the
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04 Jun 2018, 05:18
04 Jun 2018, 05:18
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Question Stats:
47% (01:34) correct
52% (01:15) wrong based on 46 sessions
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30^20 – 20^20 is divisible by all of the following values, EXCEPT:
A) 10
B) 25
C) 40
D) 60
E) 64
A) 10
B) 25
C) 40
D) 60
E) 64
Re: 30^20 – 20^20 is divisible by all of the
[#permalink]
04 Jun 2018, 05:52
04 Jun 2018, 05:52
2
(30^20 - 20^20)
=10^20 ( 3^20 - 2^20)
Clearly, divisible by 10.
Can be written as :
= 1000 * 10^17 ( 3^20 - 2^20)
Clearly, divisible by 25 & 40 ( 1000 is divisible by 25 & 40 )
Can also be written as :
= 2^6 * 5^6 * 10^14 ( 3^20 - 2^20)
Clearly, divisible by 64 ( 2^6 is divisible by 64)
Thus, eliminating options a,b,c & e, option D remains.
Hence, D.
=10^20 ( 3^20 - 2^20)
Clearly, divisible by 10.
Can be written as :
= 1000 * 10^17 ( 3^20 - 2^20)
Clearly, divisible by 25 & 40 ( 1000 is divisible by 25 & 40 )
Can also be written as :
= 2^6 * 5^6 * 10^14 ( 3^20 - 2^20)
Clearly, divisible by 64 ( 2^6 is divisible by 64)
Thus, eliminating options a,b,c & e, option D remains.
Hence, D.
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: 30^20 – 20^20 is divisible by all of the
[#permalink]
05 Jun 2018, 09:02
05 Jun 2018, 09:02
2
GreenlightTestPrep wrote:
30^20 – 20^20 is divisible by all of the following values, EXCEPT:
A) 10
B) 25
C) 40
D) 60
E) 64
A) 10
B) 25
C) 40
D) 60
E) 64
Here are some useful divisibility rules:
1. If integers A and B are each divisible by integer k, then (A + B) is divisible by k
2. If integers A and B are each divisible by integer k, then (A - B) is divisible by k
3. If integer A is divisible by integer k, BUT integer B is NOT divisible by integer k, then (A + B) is NOT divisible by k
4. If integer A is divisible by integer k, BUT integer B is NOT divisible by integer k, then (A - B) is NOT divisible by k
Now let's check the answer choices....
A) 10
30^20 = (10^20)(3^20) = (10)(10^19)(3^20), so 30^20 is divisible by 10
20^20 = (10^20)(2^20) = (10)(10^19)(2^20), so 20^20 is divisible by 10
So, by rule #2, 30^20 – 20^20 MUST be divisible by 10
ELIMINATE A
B) 25
30^20 = (5^20)(6^20) = (5^2)(5^18)(6^20) = (25)(5^18)(6^20), so 30^20 is divisible by 25
20^20 = (5^20)(4^20) = (5^2)(5^18)(4^20) = (25)(5^18)(4^20), so 20^20 is divisible by 25
So, by rule #2, 30^20 – 20^20 MUST be divisible by 25
ELIMINATE B
C) 40
30^20 = (10^20)(4^20) = (10)(10^19)(4)(4^19) = (40)(10^19)(4^19), so 30^20 is divisible by 40
20^20 = (10^20)(2^20) = (10)(10^19)(2^2)(2^18) = (40)(10^19)(2^18), so 20^20 is divisible by 40
So, by rule #2, 30^20 – 20^20 MUST be divisible by 40
ELIMINATE C
D) 60
30^20 = (30^1)(30^19) = (30^1)(2^19)(15^19) = (30)(2)(2^18)(15^19) = (60)(2^18)(15^19), so 30^20 is divisible by 60
20^20 = (5^20)(4^20) = (5^20)(2^20)(2^20). This tells us that the prime factorization of 20^20 does not have any 3's, which means 20^20 is NOT divisible by 3. And, if 20^20 is not divisible by 3, then 20^20 is NOT divisible by 60
So, by rule #4, 30^20 – 20^20 IS NOT divisible by 60
Answer: D
Cheers,
Brent
