2024 AMC 10A Problems/Problem 1

Problem

What is the value of $9901\cdot101-99\cdot10101?$

$\textbf{(A)}~2\qquad\textbf{(B)}~20\qquad\textbf{(C)}~200\qquad\textbf{(D)}~202\qquad\textbf{(E)}~2020$

Solution 1 (Direct Computation)

The likely fastest method will be direct computation. $9901\cdot101$ evaluates to $1000001$ and $99\cdot10101$ evaluates to $999999$ . The difference is $\boxed{\textbf{(A) }2}.$

Solution by juwushu.

Solution 2 (Distributive Property)

We have $\begin{align*} 9901\cdot101-99\cdot10101 &= (10000-99)\cdot101-99\cdot(10000+101) \\ &= 10000\cdot101-99\cdot101-99\cdot10000-99\cdot101 \\ &= (10000\cdot101-99\cdot10000)-2\cdot(99\cdot101) \\ &= 2\cdot10000-2\cdot9999 \\ &= \boxed{\textbf{(A) }2}. \end{align*}$ ~MRENTHUSIASM

Solution 3 (A way to get rid of some answer choices(Not recommended on the actual competition)

We simply look at the units digit of the problem we have(or take mod 10) $9901\cdot101-99\cdot10101 \equiv 1\cdot1 - 9\cdot1 = 2 \mod{10}.$ Since the only answer with 2 in the units digit is $\textbf{(A)}$ or $\textbf{(D)}$ We can then continue if you are desperate to use guess and check or a actually valid method to find the answer is $\boxed{\textbf{(A)}}$ ~mathkiddus

2024 AMC 10A (Problems • Answer Key • Resources)
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2024 AMC 10A Problems/Problem 1

Contents

Problem

Solution 1 (Direct Computation)

Solution 2 (Distributive Property)

Solution 3 (A way to get rid of some answer choices(Not recommended on the actual competition)

See also