Around the World in 80 Questions (Day 9): A number is selected at

Question

A number is selected at random from four-digit positive integers that have all their digits as even numbers. What is the probability that the selected number is divisible by 8?

(A) 1/8
(B) 3/16
(C) 1/4
(D) 8/25
(E) 2/5

vedha0 · Accepted Answer

answer is D.

total number of even numbers possible = 4*5*5*5 (since thousands digit has only 4 possiblities - 2,4,6,8)

for a number to be divisible by 8, last 3 digits together should be divisible by 8. for example - 1168 is divisible by 8 since 168 is divisible by 8

now numbers having 000,008,024,040,048,064,080,088,200,208,224... as last 3 digits are divisible by 8. note that from 200 the cycle repeats with having 2 in the hundreds digit in place of 0. 
from 000 to 088 - its 8 numbers in the range 000-199. another 8 numbers in the range 200-399. so on and so forth till 800-999. 
So the number of even numbers in 1 set of 1000 consecutive numbers from x000 to x9999 (where x can 2 4 6 or 8) is 8*5 = 40. x can have 4 possibilities. so no of favorable outcomes is 4*40

required probability is (4*40)/(4*5*5*5) = 8/25. therefore answer is D.

ankitgoswami · Answer

Number of 4-digit integers that have all their digits as even numbers = 4 * 5 * 5 * 5

For a number to be divisible by 8, its last three digits should be divisible by 8.
So lets calculate three digit numbers consisting of only even numbers divisible by 8. Include the numbers starting with 0 such as 024, 048 etc.

Number of such possibilities = 8 * 5 * 4

Probability =  8/25

Ans  D

Kinshook · Answer

Given: A number is selected at random from four-digit positive integers that have all their digits as even numbers. 
Asked: What is the probability that the selected number is divisible by 8?

Total numbers with all their digits as even numbers = {2000, 2002, 2004, 2006, 2008, 2020, 2022, 2024, 2026, 2028, 2040, 2042, 2044, 2046, 2048, 2060, 2062, 2064, 2066, 2068, 2080, 2082, 2084, 2086, 2088, ...} = 25*20
Total number with all their digits as even numbers and divisible by 8 = {2000, 2008, 2024,2040, 2048, 2064, 2080, 2088, ...} = 8*20

Total 4-digit numbers with all their digits as even numbers = 4*5*5*5 = 500 = 25*20

The probability that the selected number is divisible by 8 = 8*20/25*20 = 8/25

IMO D

UtkarshAnand · Answer

A number is selected at random from four-digit positive integers that have all their digits as even numbers. What is the probability that the selected number is divisible by 8?

(A) 1/8
(B) 3/16
(C) 1/4
(D) 8/25
(E) 2/5

Solution:

Denominator: 
All are four digit even nos. So the four places can be filled by 0,2,4,6,8.
_ _ _ _ - These 4 places can be filled in  4*5*5*5  ways.

Numerator: 
In 1 hundred set, we have 8 multiples of 8 that are purely made of even nos. - for eg: 000, 008, 024, 040, 048, 064, 080, 088. 
These 8 will repeat alternately in numbers greater than 2000, 4000, 6000, 8000. 
In numbers greater than 2000 we have total 8*5 such nos.
So, total will be  8*5*4  (to include all nos. greater than 2000, 4000, 6000, 8000).

Numerator/Denominator = 8*5*4/4*5*5*5 = 8/25.  ANSWER D

missionmba2025 · Answer

A tough one !

Number of four-digit positive integers that have all their digits as even numbers = 4 * 5 * 5 * 5

The thousands place can be filled in 4 ways (2 / 4/ 6 / 8), and all other places can be filled in 5 ways (0/2/4/6/8)

A number is divisible by 8 when the last three digits are divisible by 8.

A B C D

1000*A + 100 * B + 10 * C + D

B can be (0/2/4/6/8) , hence 100*B is always divisible by 8.

We need to concentrate only on the last two digits

00 
02
04
06
 08

20
22
 24 
26
28

40 
42
44
46
 48

60
62
 64 
66
68

80 
82
84
86
 88

8 applicable numbers.

Favorable numbers :

The last two digits can be filled in 8 ways , the thousands place can be filled in 4 ways and the hundred place can be filled in 5 ways

Probability = (4 * 5 * 8) / (4 * 5 * 5 * 5) = 8 / 25

IMO D

NoobGmatTaker · Answer

No of 4 digit no: 4*5*5*5 = 500

for a digit to be divisible by 8, last three digit should be divisible by 8.
Possible 3 digit till 100:   * 5 as these number will repeat with 0,2,4,6,8 =>. Total values = 8*5 = 40
The 4th digit can be taken in 4 ways(except 0). so total ways =. 40*4 =160

P(4 digit number using even digit to be divisible by 8 =  \frac{160}{500} => \frac{8}{25}

Correct answer is D

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