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

Latest revision as of 16:25, 11 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 $\boxed{4}$ .

~Bradygho

Solution 2

By multiplying out $16 \cdot 25$ , $161 \cdot 252$ , and $1616 \cdot 2525$ , we notice that the first $2$ digits don't change even when we continue to add more digits. With this observation, we can conclude that the first digit of the product is $\boxed{4}$ .

~Mathdreams

Solution 3

Remove factors of $16$ and $25$ to get $\left(\underbrace{101010101 \cdots}_{\text{50 0s and 50 1s}} \right)^2 \cdot 400$ . Recall by Pascal's triangle that $11=121$ , $101=10201$ , so the leftmost digit is guaranteed to be $1$ . Now, multiplying by our scale factor the answer is $\boxed{4}$ $\linebreak$ ~Geometry285

@@ Line 3: / Line 3: @@
 ==Solution==
-asdf
+We notice that
+<cmath>16000\cdots \times 25000\cdots = 16 \times 25 \times 10^{198} = 400 * 10^{198}</cmath>
+In addition, we notice that
+<cmath>16200\cdots \times 25300\cdots = 162 \times 253 \times 10^{194} = 40986 \times 10^{194}</cmath>
+Since
+<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>
+We conclude that the leftmost digit must be <math>\boxed{4}</math>.
+~Bradygho
+==Solution 2==
+By multiplying out <math>16 \cdot 25</math>, <math>161 \cdot 252</math>, and <math>1616 \cdot 2525</math>, we notice that the first <math>2</math> digits don't change even when we continue to add more digits. With this observation, we can conclude that the first digit of the product is <math>\boxed{4}</math>.
+~Mathdreams
+==Solution 3==
+Remove factors of <math>16</math> and <math>25</math> to get <math>\left(\underbrace{101010101 \cdots}_{\text{50 0s and 50 1s}} \right)^2 \cdot 400</math>. Recall by Pascal's triangle that <math>11=121</math>, <math>101=10201</math>, so the leftmost digit is guaranteed to be <math>1</math>. Now, multiplying by our scale factor the answer is <math>\boxed{4}</math>
+<math>\linebreak</math>
+~Geometry285
+==See also==
+#[[2021 JMPSC Accuracy Problems|Other 2021 JMPSC Accuracy Problems]]
+#[[2021 JMPSC Accuracy Answer Key|2021 JMPSC Accuracy Answer Key]]
+#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
+{{JMPSC Notice}}

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