What is (100,016)(99,984)?

Solution:

We see that 100,016 x 99,984 will have a units digit of 4.

Since 10^10 has a units digit of 0, we see that answer choice E ends in a zero, so we do not have to test that answer.

We also see that answers A - D, have some power of 10, which ends in zero, minus an integer.

The way for a number ending in zero to subtract to get a units digit of 4 is to subtract another number ending in a 6.

Since 2^8 = 256, we see that 10^10 - 2^8 will end in a 4.

Alternate solution:

(100,016)(99,984)

(100,000 + 16)(100,000 - 16)

(10^5 + 2^4)(10^5 - 2^4)

10^10 - 2^8

Answer: C

\((100,016)(99,984) = (100,000 + 16)(100,000 - 16) = 10^{10} - 4^2 = 10^{10} - 2^8\) => Answer (C)

\((100,016)(99,984)\)

\(= (100,016)(99,984)\)

\(= ( 100,000 + 16 )( 100,000 - 16 )\)

\(= ( 100,000^2 - 16^2 )\)

\(= ( 10^{10} - 2^8 )\) , Hence Answer must be (C)

Clearly, the GMAT does not expect you to actually multiply the given numbers to get to the answer. Even if you do, you will have a hard time matching your answer to the expressions given in the answer options.

Upon close observation, both the numbers are close to one common number, which is 100,000.

100,016 = 100,000 + 16 and 99,984 = 100,000 – 16

Therefore, the given product can be written in the form of (a-b) (a+b), after which we apply standard algebraic identities to get to the final answer.

(100,016) (99,984) = (100,000 + 16) (100,000 – 16) = \((100,000)^2\) – \((16)^2\), since (a-b) (a+b) = \(a^2\)– \(b^2\)

100,000 = \(10^5\); therefore, \((100,000)^2\) = \((10^5)^2\) = \(10^{10}\)
16 = \(2^4\); therefore, \((16)^2\) = \((2^4)^2\) = \(2^8\)

Therefore, the required answer is \(10^{10}\) – \(2^8\)

The correct answer option is C.

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