Problem Solving

30^20 - 20^20 is divisible by all of the following values, EXCEPT:

A) 10
B) 25
C) 40
D) 60
E) 64

Answer: D
Difficulty level: 600 - 650
Source: www.gmatprepnow.com

Brent@GMATPrepNow wrote:30^20 - 20^20 is divisible by all of the following values, EXCEPT:

A) 10
B) 25
C) 40
D) 60
E) 64

Answer: D
Difficulty level: 600 - 650
Source: www.gmatprepnow.com

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