30^20 – 20^20 is divisible by all of the following

Let's simplify the given exponents:

\(30^{20} – 20^{20}\)
=\(10^{20}\)*\(3^{20}\)-\({10}^{20}\)*\(2^{20}\)
=\(10^{20}\)(\(3^{20}\)-\(2^{20}\))

For divisibility, lets check with the options:-

A .\(10^{20}\) is divisible by 10
B. \(10^{20}\)=100*\(10^{18}\) is divisible by 25
C. \(10^{20}\)=(\(2^3\)*5)*\(2^{17}\)*\(5^{19}\) is divisible by 40
D. \(10^{20}\)=\(2^{20}\)*\(5^{20}\) is not divisible by \(2^2\)*5*3
E. \(10^{20}\)=\(2^6\)*\(2^{14}\)*\(5^{20}\) is divisible by 64

Hence the correct answer is option (D)

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)(3^20) = (10^3)(10^17)(3^20) = = (1000)(10^17)(3^20) = (40)(25)(10^17)(3^20), 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

30^20 – 20^20 = 2^20 * 5^20 (3^20- 2^20) = 10^20 (3^20 - 2^20)

Prime factorizing the options:

A) 10 = 2 * 5
B) 25= 5 * 5
C) 40 = 2^3 * 5
D) 60 = 2^2 * 3 * 5
E) 64 = 2^6

Option D has 3 as Prime factor. So, 60 does not divide (3^20- 2^20).

Correct Answer- D

\(30^{20} – 20^{20}\)
=\(10^{20}\)*\(3^{20}\)-\({10}^{20}\)*\(2^{20}\)
=\(10^{20}\)(\(3^{20}\)-\(2^{20}\))
=\(5^{20}\)\(2^{20}\)(\(3^{20}\)-\(2^{20}\))

A) 10 = 2 * 5. divisible by this value.
B) 25 = \(5^2\) . Clearly divisible by this value.
C) 40 = \(2^3 * 5\). divisible by this value.
D) 60 = \(2^2 *3 * 5\). NOT divisible by this value.
E) 64 = \(2^6\). divisible by this value.

Ans - D.

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