Which of the following is equivalent to 1,000,001

Question

Which of the following is equivalent to  \frac{1,000,001 	imes 999,999}{1001} ?

A)  1,001 	imes 999,999 
B)  10,001 	imes 99,999 
C)  100,001 	imes 9,999 
D)  1,000,001 	imes 999 
E)  10,000,001 	imes 99

ArunSharma12 · Accepted Answer

\frac{1,000,001 	imes 999,999}{1001}  =  \frac{(10^6 + 1) 	imes (10^6 - 1)}{(10^3+1)}  =  \frac{(10^6 + 1) 	imes (10^3 + 1) 	imes (10^3 - 1)}{(10^3+1)}  =  (10^6 + 1) 	imes (10^3 - 1)  =  (10^6 + 1) 	imes 999 
Ans: D

BrentGMATPrepNow · Answer

\frac{1,000,001 	imes 999,999}{1,001}=\frac{(1,000,000 + 1)(1,000,000 - 1)}{1,001}

=\frac{(1,000,000 + 1)(1,000^2 - 1^2)}{1,001}

=\frac{(1,000,001)(1,000+ 1)(1,000-1)}{1,001}

=\frac{(1,000,001)(1,001)(999)}{1,001}

=1,000,001 	imes 999

Answer: D

Cheers, 
Brent

MartyMurray · Answer

Which of the following is equivalent to  \frac{1,000,001 	imes 999,999}{1001} ?

We can do this one quickly if we notice that 999,999 = 999 × 1001.

So,  \frac{1,000,001 	imes 999,999}{1001} = 1,000,001× 999 .

A)  1,001 	imes 999,999 
B)  10,001 	imes 99,999 
C)  100,001 	imes 9,999 
D)  1,000,001 	imes 999 
E)  10,000,001 	imes 99

Correct answer:   D

egmat · Answer

The expression is ( 1,000,001  ×  999,999 ) /  1001 .

The key question is: can we simplify by dividing  1001  into either  1,000,001  or  999,999 ?

Let's try  999,999  ÷  1001 .

Here's a neat trick:  1001  ×  1000  =  1,001,000 . That's just slightly more than  999,999 . Specifically:
 1,001,000  -  1,001  =  999,999 
So  1001  ×  1000  -  1001  ×  1  =  999,999 
Which means  1001  ×  999  =  999,999 . ✓

Now the original expression becomes:
( 1,000,001  ×  999,999 ) /  1001  =  1,000,001  × ( 999,999  /  1001 ) =  1,000,001  ×  999

Answer: D

The common mistake here is trying to divide  1001  into  1,000,001  instead.  If you do that, you get  1,000,001  /  1001  ≈  999.001 ..., which isn't a clean number. So  1001  doesn't divide evenly into  1,000,001  — it divides into  999,999 .

General principle:  When you see numbers like  999,999  or  1,000,001 , think about how they relate to nearby round numbers.   999,999  =  1,000,000  -  1 . And  1001  ×  999  =  1001  × ( 1000  -  1 ) =  1,001,000  -  1,001  =  999,999 . Recognizing these patterns is the fastest path to the answer.

Which of the following is equivalent to 1,000,001

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Which of the following is equivalent to 1,000,001

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