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Difficulty:
35%
(medium)
Question Stats:
76% (01:40) correct
24%
(02:14)
wrong
based on 106
sessions
History
Date
Time
Result
Not Attempted Yet
How many integers between 100 and 1,000 have a hundreds digit that is even, a tens digit that is odd, and a non-zero units digit that is divisible by 3?
A. 15
B. 30
C. 45
D. 60
E. 75
_________________
A. 15
B. 30
C. 45
D. 60
E. 75
_________________
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Bunuel
Solution:
- We are looking for integers between 100 and 100 such that
- Hundred digit is even. This means 200s, 400s, 600s and 800s. 4 options for the hundredth-digit place
- Tens digit is odd. This means (210s, 230s, 250s, 270s, 290s), (410s, 430s, 450s, 470s, 490s), (610s, 630s,...), .....870s and 890s. 5 options for the tenth-digit place
- Unit digit is non zero and the integer is divisible by 3. This means (213, 216, 219), (231, 234, 237), (252, 255, 258). 3 options for the unit digit
- So, total number of such integers \(=4\times 5\times 3=60\)
Hence the right answer is Option D
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total 3 digit starting with even number between 100 and 1000 = 400
now with ten digit as odd
we will be left with 200
now 3 rd digit is non zero so it cannot
be (210,230,250,270,290)
total 20
subtracting we get 180 numbers left
now unit digit divisible by 3 means it can be 3,6,9
213,216,219
233,236,239
253,256,259
273,276,279
293,296,299
total 15 *4 =60 is right ans
now with ten digit as odd
we will be left with 200
now 3 rd digit is non zero so it cannot
be (210,230,250,270,290)
total 20
subtracting we get 180 numbers left
now unit digit divisible by 3 means it can be 3,6,9
213,216,219
233,236,239
253,256,259
273,276,279
293,296,299
total 15 *4 =60 is right ans
Posts: 477
Deconstructing the Question
We need to find the number of 3-digit integers (between 100 and 1,000) that satisfy three specific conditions for each digit.
1. Hundreds Digit (Must be Even)
The hundreds digit of a 3-digit number cannot be 0.
Possible even digits: {2, 4, 6, 8}
Total options = 4
2. Tens Digit (Must be Odd)
Possible odd digits: {1, 3, 5, 7, 9}
Total options = 5
3. Units Digit (Non-zero and divisible by 3)
Digits divisible by 3 are {0, 3, 6, 9}.
Since it must be "non-zero", the possible digits are: {3, 6, 9}
Total options = 3
4. Total Combination
Using the Fundamental Counting Principle, we multiply the number of options for each position:
\(Total = 4 * 5 * 3 = \) 60
The correct answer is D.
We need to find the number of 3-digit integers (between 100 and 1,000) that satisfy three specific conditions for each digit.
1. Hundreds Digit (Must be Even)
The hundreds digit of a 3-digit number cannot be 0.
Possible even digits: {2, 4, 6, 8}
Total options = 4
2. Tens Digit (Must be Odd)
Possible odd digits: {1, 3, 5, 7, 9}
Total options = 5
3. Units Digit (Non-zero and divisible by 3)
Digits divisible by 3 are {0, 3, 6, 9}.
Since it must be "non-zero", the possible digits are: {3, 6, 9}
Total options = 3
4. Total Combination
Using the Fundamental Counting Principle, we multiply the number of options for each position:
\(Total = 4 * 5 * 3 = \) 60
The correct answer is D.
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