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avatar+191 

1.Among all fractions $x$ that have a positive integer numerator and denominator and satisfy $$\frac{9}{11} \le x \le \frac{11}{13},$$ which fraction has the smallest denominator?

 

2. 

What is the largest 4-digit number that is equal to the cube of the sum of its digits?

 Jun 22, 2021
 #1
avatar+121048 
+2

1.   

 

9/11  ≤ x  ≤ 11/13

 

.8181 ...  ≤  x  ≤  .846153  ......

 

x =  5/6   =  .8333......

 

 

cool cool cool

 Jun 22, 2021
 #2
avatar+121048 
+2

2.

 

5832  =   

 

5 + 3 + 8 +  2   =   18

 

 

18^3  =  5832

 

 

cool cool cool

 Jun 22, 2021
 #3
avatar+2250 
+1

Nice answer, but how did you solve it?

 

I tried putting it in this form, but I don't know where to go from there. 

1000a + 100b + 10c + d = (a+b+c+d)^3

 

=^._.^=

catmg  Jun 22, 2021
 #4
avatar+121048 
+1

Just  a little  trial and error, catng

 

21^3   = 9261      ⇒  sum =  18        

20^3  = 8000       ⇒  sum = 8

19^3  = 6859       ⇒  sum   = 28

18^3  =  5832     ⇒  sum  = 18

 

cool cool cool

CPhill  Jun 23, 2021
 #5
avatar+2250 
0

Smart. :)

Surprisingly, cbrt(9999) isn't that big, I assumed there were a lot of cases. 

 

=^._.^=

catmg  Jun 23, 2021

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