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# Help

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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
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1.

9/11  ≤ x  ≤ 11/13

.8181 ...  ≤  x  ≤  .846153  ......

x =  5/6   =  .8333......   Jun 22, 2021
#2
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2.

5832  =

5 + 3 + 8 +  2   =   18

18^3  =  5832   Jun 22, 2021
#3
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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
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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   CPhill  Jun 23, 2021
#5
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Smart. :)

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

=^._.^=

catmg  Jun 23, 2021