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

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06 Jan 2015, 09:17

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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

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).
_________________

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

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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)

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

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23 Aug 2016, 10:37

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

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31 Oct 2016, 02:19

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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

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

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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