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

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 kNow let's check the answer choices....

A) 1030^20 = (10^20)(3^20) = (

10)(10^19)(3^20), so

30^20 is divisible by 1020^20 = (10^20)(2^20) = (

10)(10^19)(2^20), so

20^20 is divisible by 10So, by

rule #2,

30^20 – 20^20 MUST be divisible by 10ELIMINATE A

B) 2530^20 = (5^20)(6^20) = (5^2)(5^18)(6^20) = (

25)(5^18)(6^20), so

30^20 is divisible by 2520^20 = (5^20)(4^20) = (5^2)(5^18)(4^20) = (

25)(5^18)(4^20), so

20^20 is divisible by 25So, by

rule #2,

30^20 – 20^20 MUST be divisible by 25ELIMINATE B

C) 4030^20 = (10^20)(4^20) = (10)(10^19)(4)(4^19) = (

40)(10^19)(4^19), so

30^20 is divisible by 4020^20 = (10^20)(2^20) = (10)(10^19)(2^2)(2^18) = (

40)(10^19)(2^18), so

20^20 is divisible by 40So, by

rule #2,

30^20 – 20^20 MUST be divisible by 40ELIMINATE C

D) 6030^20 = (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 6020^20 = (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 60So, by

rule #4,

30^20 – 20^20 IS NOT divisible by 60Answer: D

Cheers,

Brent

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Brent Hanneson – Creator of greenlighttestprep.com

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