Difference between revisions of "2021 JMPSC Accuracy Problems/Problem 14"
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− | == Solution 2 == | + | ==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>4</math>. | + | 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 | ~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}} |
Latest revision as of 17:25, 11 July 2021
Problem
What is the leftmost digit of the product
Solution
We notice that
In addition, we notice that
Since
We conclude that the leftmost digit must be .
~Bradygho
Solution 2
By multiplying out ,
, and
, we notice that the first
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
.
~Mathdreams
Solution 3
Remove factors of and
to get
. Recall by Pascal's triangle that
,
, so the leftmost digit is guaranteed to be
. Now, multiplying by our scale factor the answer is
~Geometry285
See also
The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.