Difference between revisions of "1960 AHSME Problems/Problem 16"

m (Problem 16)
m (Solution)
Line 12: Line 12:
 
<cmath>69 = 2 \cdot 25 + 19</cmath>
 
<cmath>69 = 2 \cdot 25 + 19</cmath>
 
Similarly, dividing <math>19</math> by <math>5</math> results in quotient of <math>3</math> and remainder of <math>4</math>, so rewrite the number as
 
Similarly, dividing <math>19</math> by <math>5</math> results in quotient of <math>3</math> and remainder of <math>4</math>, so rewrite the number as
<cmath>69 = 2 \cdot 25 + 3 \cdot 5 + 4 \cdot 1</cmath>.
+
<cmath>69 = 2 \cdot 25 + 3 \cdot 5 + 4 \cdot 1</cmath>
 
Thus, the number in base <math>5</math> can be written as <math>234_5</math>, so the answer is <math>\boxed{\textbf{(C)}}</math>
 
Thus, the number in base <math>5</math> can be written as <math>234_5</math>, so the answer is <math>\boxed{\textbf{(C)}}</math>
 
 
  
 
==See Also==
 
==See Also==
 
{{AHSME 40p box|year=1960|num-b=15|num-a=17}}
 
{{AHSME 40p box|year=1960|num-b=15|num-a=17}}

Revision as of 11:03, 11 May 2018

Problem

In the numeration system with base $5$, counting is as follows: $1, 2, 3, 4, 10, 11, 12, 13, 14, 20,\ldots$. The number whose description in the decimal system is $69$, when described in the base $5$ system, is a number with:

$\textbf{(A)}\ \text{two consecutive digits} \qquad\textbf{(B)}\ \text{two non-consecutive digits} \qquad \\ \textbf{(C)}\ \text{three consecutive digits} \qquad\textbf{(D)}\ \text{three non-consecutive digits} \qquad \\ \textbf{(E)}\ \text{four digits}$

Solution

Since $25<69<125$, divide $69$ by $25$. The quotient is $2$ and the remainder is $19$, so rewrite the number as \[69 = 2 \cdot 25 + 19\] Similarly, dividing $19$ by $5$ results in quotient of $3$ and remainder of $4$, so rewrite the number as \[69 = 2 \cdot 25 + 3 \cdot 5 + 4 \cdot 1\] Thus, the number in base $5$ can be written as $234_5$, so the answer is $\boxed{\textbf{(C)}}$

See Also

1960 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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 31 32 33 34 35 36 37 38 39 40
All AHSME Problems and Solutions
Invalid username
Login to AoPS