Difference between revisions of "2021 JMPSC Accuracy Problems/Problem 14"

(Created page with "==Problem== What is the leftmost digit of the product <cmath>\underbrace{161616 \cdots 16}_{100 \text{ digits }} \times \underbrace{252525 \cdots 25}_{100 \text{ digits }}?</c...")
 
(Solution)
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==Solution==
 
==Solution==
asdf
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We notice that
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<cmath>16000\cdots \times 25000\cdots = 16 \times 25 \times 10^{198} = 400 * 10^{198}</cmath>
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In addition, we notice that
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<cmath>16200\cdots \times 25300\cdots = 162 \times 253 \times 10^{194} = 40986 \times 10^{194}</cmath>
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Since
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<cmath>16000\cdots \times 25000\cdots < \underbrace{161616 \cdots 16}_{100 \text{ digits }} \times \underbrace{252525 \cdots 25}_{100 \text{ digits }} < 16200\cdots \times 25300\cdots</cmath>
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We conclude that the leftmost digit must be <math>\boxed{4}</math>.

Revision as of 22:52, 10 July 2021

Problem

What is the leftmost digit of the product \[\underbrace{161616 \cdots 16}_{100 \text{ digits }} \times \underbrace{252525 \cdots 25}_{100 \text{ digits }}?\]

Solution

We notice that \[16000\cdots \times 25000\cdots = 16 \times 25 \times 10^{198} = 400 * 10^{198}\] In addition, we notice that \[16200\cdots \times 25300\cdots = 162 \times 253 \times 10^{194} = 40986 \times 10^{194}\]

Since \[16000\cdots \times 25000\cdots < \underbrace{161616 \cdots 16}_{100 \text{ digits }} \times \underbrace{252525 \cdots 25}_{100 \text{ digits }} < 16200\cdots \times 25300\cdots\]

We conclude that the leftmost digit must be $\boxed{4}$.

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