If 1

Question

If 1 <= x <= 100, what fraction of values for integer x have the same units digit as x^3?

A) 1/10
B) 1/5
C) 2/5
D) 1/2
E) 3/5

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ManifestDreamMBA · Answer

Numbers with same units digit in their cubes: 1,4,5,6,9,0
So from 1 to 10, there are 6 such numbers
From 1-100, we will have 6*10 = 60 such cases
Fractions of values for integer x have the same units digit as x^3 = 60/100 = 3/5

Whoisdmx15 · Answer

Checking for which integers will have same units digits as cubes:

Cube of units digit of integer x   
  Units digit of Cube   
 
    1^3    
  1  
 
   2^3   
 8 
 
   3^3  
 7 
 
    4^3    
  4  
 
    5^3    
  5  
 
    6^3    
  6  
 
   7^3   
 3 
 
   8^3   
 2 
 
    9^3    
  9  
 
    0^3    
  0

Number of integers with units digits 1 = 10, 4 = 10, 5 = 10, 6 = 10 , 9 = 10, 0 = 10 = total 60
Hence total fraction required = 60/100 = 3/5

IMO Ans E

GMATinsight · Answer

x and x^3 will have the same digit for the number which have unit digit whose cyclicity is either 2 or 1

Cyclicity is the cycles of powers after which the unit digit repeats i.e. Cyclicity of 2 means that unit digit repeats after two higher powers

Cyclicity of 1: The numbers with unit digit {0, 1, 5, 6} 
 Cyclicity of 2: The numbers with unit digit {4, 9} 
 Cyclicity of 4: The numbers with unit digit {2, 3, 7, 8}

Numbers from 1 to 10 that have cyclicity 1 or 2 = {0, 1, 4, 5, 6, 9} i.e. 6 numbers

Every 10 consecutive numbers in such sequence will have 6 numbers that will have cyclicity of 1 or 2

hence, total numbers for which x and x^3 will have same unit digit int he given range = 6*10 = 60

Probability of choosing a number of this type = 60/100 = 3/5

Answer: Option E

Related Video:

https://www.youtube.com/watch?v=j5F_LevQd-Q

giuseppemvn · Answer

At first I didn't understand the "fraction of values" part, then I figured it out:

Count of all possible values that keep the same unit digit when elevated to the third power/Total count of possible values.

Total number of possible values for   x  :  100. This will be our denominator.

Number of values that keep the unit digit when elevated to the third power:  all the numbers that end in {0,1,4,5,6,9} = 6 numbers * 10 different possible ten digits = 60.

\frac{60}{100}=\frac{3}{5}

Answer E .

Cube of units digit of integer x	Units digit of Cube
\(1^3\)	1
\(2^3\)	8
\(3^3\)	7
\(4^3\)	4
\(5^3\)	5
\(6^3\)	6
\(7^3\)	3
\(8^3\)	2
\(9^3\)	9
\(0^3\)	0

If 1<=x<=100, what fraction of values for integer x have the same unit

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