Problem:
What is the value of 101⋅9,901−99⋅10,101 ?
Answer Choices:
A. 2
B. 20
C. 21
D. 200
E. 2020
Solution:
Write the difference as
(100+1)⋅(9900+1)−99⋅(10,000+100+1).
Applying the distributive property gives
(990,000+9,900+100+1)−(990,000+9,900+99)=100+1−99=(A)2.
OR
Let x=100. Then the minuend (the first quantity in the subtraction operation) is
(x+1)(x2−x+1)=x3+1,
and the subtrahend (the quantity being subtracted from the minuend) is
(x−1)(x2+x+1)=x3−1.
The difference is
(x3+1)−(x3−1)=(A)2.
The problems on this page are the property of the MAA's American Mathematics Competitions