1984 AHSME Problems/Problem 5: Difference between revisions
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==Problem 5== | ==Problem 5== | ||
The largest integer <math> n </math> for which <math> n^{200}<5^{300} </math> is | The largest [[integer]] <math> n </math> for which <math> n^{200}<5^{300} </math> is | ||
<math> \mathrm{(A) \ }8 \qquad \mathrm{(B) \ }9 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ }11 \qquad \mathrm{(E) \ } 12 </math> | <math> \mathrm{(A) \ }8 \qquad \mathrm{(B) \ }9 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ }11 \qquad \mathrm{(E) \ } 12 </math> | ||
==Solution== | ==Solution== | ||
Since both sides are positive, we can take the <math> 100th </math> root of both sides to find the largest integer <math> n </math> such that <math> n^2<5^3 </math>. Fortunately, this is simple to evaluate: <math> 5^3=125 </math>, and the largest square less than <math> 125 </math> is <math> 11^2=121 </math>, so the largest <math> n </math> is <math> 11, \boxed{\text{D}} </math>. | Since both sides are positive, we can take the <math> 100th </math> root of both sides to find the largest integer <math> n </math> such that <math> n^2<5^3 </math>. Fortunately, this is simple to evaluate: <math> 5^3=125 </math>, and the largest [[Perfect square|square]] less than <math> 125 </math> is <math> 11^2=121 </math>, so the largest <math> n </math> is <math> 11, \boxed{\text{D}} </math>. | ||
==See Also== | ==See Also== | ||
{{AHSME box|year=1984|num-b=4|num-a=6}} | {{AHSME box|year=1984|num-b=4|num-a=6}} | ||
{{MAA Notice} | |||
Latest revision as of 05:08, 23 February 2023
Problem 5
The largest integer 
for which 
is
Solution
Since both sides are positive, we can take the 
root of both sides to find the largest integer such that 
. Fortunately, this is simple to evaluate: 
, and the largest square less than 
is 
, so the largest is 
.
See Also
| 1984 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
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