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
_________________
Brent Hanneson - founder of Greenlight Test Prep