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How many positive even integers less than 100 contain digits 4 or 7?

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How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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Tough and Tricky questions: Number Properties.



How many positive even integers less than 100 contain digits 4 or 7?

A. 16
B. 17
C. 18
D. 19
E. 20

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[Reveal] Spoiler: OA

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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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One digit numbers: 4
Two digit numbers:
4 at tens place: 40,42,44,46,48
7 at tens place: 70,72,74,76,78
4 at units place: 14,24,34,54,64,84,94 (duplicates removed)
If 7 is at units place, the number cant be even
Total:1+5+5+7= 18

Answer C

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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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New post 06 Jan 2015, 09:41
Bunuel wrote:

Tough and Tricky questions: Number Properties.



How many positive even integers less than 100 contain digits 4 or 7?

A. 16
B. 17
C. 18
D. 19
E. 20

Kudos for a correct solution.


Number of Even integers contain 7 = 5(70,72,74,76,78)
Number of Even integers contain 4 = 13(40,42,44,46,48, 4,14,24,34,54,64,84,94)

18

Ans C
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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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New post 07 Jan 2015, 01:04
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Answer = C = 18

"4" is even... so it will be found in units as well as ten's place

"7" is odd.... so it will be found in only ten's place

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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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New post 08 Jan 2015, 03:32
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Here are the numbers

04, 14, 24, 34, 44, 54, 64, 74, 84, 94
70, 72, 74, 76, 78
40, 42, 44, 46, 48

18 is the answer

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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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New post 08 Jan 2015, 03:41
Bunuel wrote:

Tough and Tricky questions: Number Properties.



How many positive even integers less than 100 contain digits 4 or 7?

A. 16
B. 17
C. 18
D. 19
E. 20

Kudos for a correct solution.


Is there any other method to solve this? instead of actually counting the numbers?

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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Number Properties.



How many positive even integers less than 100 contain digits 4 or 7?

A. 16
B. 17
C. 18
D. 19
E. 20

Kudos for a correct solution.


OFFICIAL SOLUTION:

(C) There are 2 one-digit numbers: 4 and 7. Only one of them is even.Among two-digit numbers (let’s denote them as ab) there are:

- numbers that have 4 or 7 as the first digit (4b, 7b).
Since even number must end by even digit b can be any of {0, 2, 4, 6 or 8}. So there are 2 × 5 = 10 such numbers.

- numbers that have 4 as the second digit (a4) are all even.
Digit a can be any except 0. So there are 1 × 9 = 9 such numbers.

- numbers that have 7 as the second digit (a4, b7) are all odd.

Note, that we’ve counted twice the two-digit numbers that have 4 as the second digit and 4 or 7 as the first (44, 74).

Therefore we have 1 + 10 + 9 – 2 = 18 possible variants overall.

The correct answer is choice (C).
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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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New post 05 Feb 2015, 07:31
0 --10 ==> 4,7 --2
11-20 ==> 14,17 --2
21-30 ==> 24,27 --2
As you can see we will have our 2 numbers in every 10 number sequence. But 40-49 and 70 -79 will have 10 each. (20)

Totol = 2 * 8 + 20 = 36

But we want only even hence 36/2 = 18

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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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New post 23 Aug 2016, 12:02
I have a question.
The problem stipulates 4 or 7, not 4 and 7.
Shouldn't 74 be excluded, leaving 17 as the correct answer?

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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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Lets solve this question using the properties of a complement.
First we’ll find out total number of even integers and subtract from this even integers that don’t have both 4 and 7.
n(4 U 7) = N(total even) – n(4 ̅⋂7 ̅)
Total number of even integers (N):
– One digit (2,4,6,8) = 4
Two digit (9 choices for tens digit – except 0; 5 choices for units digit – even including 0) = 9*5=45
Total 45+4=49
Number of even integers which do not have both 4 and 7:
One digit (2,6,8) = 3
Two digit (7 choices for tens digit excluding 0, 4 and 7; 4 choices for units digit excluding 4) = 7*4 = 28
Total 28+3=31
Number of even integers which have either 4 or 7 = 49-31=18

Answer C.

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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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New post 18 Nov 2016, 21:27
0-9 --> 4
10-19 --> 14
20-29 --> 24
30-39 --> 34
40-49 --> 40,42,44,46,48
50-59 --> 54
60-69 --> 64
70-79 --> 70,72,74,76,78
80-89 --> 84
90-99 --> 94

18 total, C.

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Re: How many positive even integers less than 100 contain digits 4 or 7? [#permalink]

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New post 12 Aug 2017, 08:07
Bunuel wrote:

Tough and Tricky questions: Number Properties.



How many positive even integers less than 100 contain digits 4 or 7?

A. 16
B. 17
C. 18
D. 19
E. 20

Kudos for a correct solution.


We will have to write the even integers to check the numbers.
So, first we will write the numbers whose unit's digit is 4.
Nos. are 04,14,24,...........,94 . Total 10 numbers. The point to be noted is that it contains 44 and 74.

Now we will check the even numbers starting with 4. They are 40,42,46,48 (44 is already considered) . Total 4

Now we will check the even numbers starting with 7. They are 70,72,76,78 (74 is already considered) . Total 4

So, total no. of positive even integers less than 100 contain digits 4 or 7 = 10 +4+4 = 18

Answer C
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Re: How many positive even integers less than 100 contain digits 4 or 7?   [#permalink] 12 Aug 2017, 08:07
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