GATE Electrical 2019 | GA | Question: 6

Question

How many integers are there between $100$ and $1000$ all of whose digits are even?

• A. $60$
• B. $80$
• C. $100$
• D. $90$ 

Comments

$\text{Between}\: 100\: \text{and}\: 1000\: \text{means}:\: 100< \text{Numbers} < 1000$

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this question is tagged under arithmetic-series on GO pdf but this is clearly a counting problem

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we can only take 200's,400's,600's and 800's and in between them only 0's,20's,40's,60's and 80's. There are only 5 even numbers in between 0 to 9 so 5*4*4=100.

Straightforward thing!!

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5*5*4=100 instead of 5*4*4=100

Answer 1

Let the 3 digit even number be ${\color{Red} \bigstar}\;{\color{Green} \bigstar}\;{\color{Blue} \bigstar}$

We can fill ${\color{Red} \bigstar}\;$ with $4$ values ($2,4,6,8$)

We can fill ${\color{Green} \bigstar}\;$ with $5$ values ($0,2,4,6,8$)

We can fill ${\color{Blue} \bigstar}\;$ with $5$ values ($0,2,4,6,8$).

So total $3$ digit even number possible = $4*5*5$ = $100$

$\therefore$ Option C $100$ is the correct answer.

Comments

$5\times5\times4 \neq 80$

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corrected...silly mistake.

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awesome approach

 

Answer 2

From 100 to 199 there is no number exist whose all digits are even (0,2,4,6,8)

From 200 to 208,220 to 228,240 to 248,260 to 268,280 to 288   = 5+5+5+5+5= 25

Same as 400 to 488 have 25 total numbers

600 to 688 have 25 total numbers

800 to 888 have 25 total numbers

Hence integers between 100 and 1000 whose digits are even =25+25+25=100

answer C.

Answer 3

_ _ _ 

1. In the units place the possible number can be vary from 0,2,4,6,8 
2. In the tens place similarly, outcome is the same as number can vary from 0 to 8
3. But the story is not the same in case of hundredth place as number should be greater than 100 thus the only possible outcome is 2,4,6,8 
4.  So 4*5*5 = 100 possibilities are there in which all the digits can be even.  

Answer 4

1. Hundreds digit: The hundreds place must be an even digit. The even digits are 0, 2, 4, 6, and 8, but since the number must be between 100 and 1000, the hundreds digit cannot be 0. Therefore, the possible digits for the hundreds place are 2, 4, 6, and 8. This gives us 4 options.

2. Tens digit: The tens place can be any even digit, including 0, 2, 4, 6, and 8. So, there are 5 options for the tens digit.

3. Ones digit: Similarly, the ones place can be any even digit, so there are 5 options for the ones digit.

Now, to find the total number of integers, multiply the number of options for each place: 100