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# How many three digit numbers are divisible by 78 or 91?

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Director
Joined: 14 Jan 2007
Posts: 686
How many three digit numbers are divisible by 78 or 91?  [#permalink]

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12 May 2007, 13:49
3
8
00:00

Difficulty:

95% (hard)

Question Stats:

44% (02:11) correct 56% (02:28) wrong based on 153 sessions

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How many three digit numbers are divisible by 78 or 91?

A. 17
B. 19
C. 20
D. 21
E. 22
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Re: How many three digit numbers are divisible by 78 or 91?  [#permalink]

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08 Jul 2014, 03:32
5
2
78 is nearer to 80

3 digit numbers divisible by 80 start from 160 to 960 (Total = 11)

91 is nearer to 90

3 digit numbers divisible by 90 start from 180 to 990 (Total = 9)

LCM of 78 & 91

= 13 * 6 * 7

= 546

There is only 1 common divisible by 78 & 91

Total three digit numbers divisible by 78 or 91

= 9 + 11 - 1

= 19

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##### General Discussion
Senior Manager
Joined: 03 May 2007
Posts: 250
Re: How many three digit numbers are divisible by 78 or 91?  [#permalink]

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12 May 2007, 14:01
1
vshaunak wrote:
How many three digit numbers are divisible by 78 or 91?

a. 17
b. 19
c. 20
d. 21
e. 22

Well there are 9 three digit numbers that are divisible by 91 (182, ....910)
And the are 11 three digit numbers that are divisible by 156.... 780, 780+78, 780+78+78)

Because we have OR we can add those two numbers

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Posts: 276
Re: How many three digit numbers are divisible by 78 or 91?  [#permalink]

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12 May 2007, 18:42
2

Explanation:

78= 2*3*13 now multiples of 78, 156 ....780, now 1000-780 = 220 only two more muktiples of 78 can exists. So total number of 3 digit Multiples of 78 are 9+2=11

91= 13*7--Total number of three digit multiples --9

No remember we have a common multiples as well-- 13*7*6=91*6=546

So total number of multiples--11+9 -1 =19.

I hope that helps!
Saurabh Malpani

vshaunak@gmail.com wrote:
How many three digit numbers are divisible by 78 or 91?

a. 17
b. 19
c. 20
d. 21
e. 22

Senior Manager
Joined: 03 May 2007
Posts: 250
Re: How many three digit numbers are divisible by 78 or 91?  [#permalink]

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12 May 2007, 18:46
Thanks saurabhmalpani i didn't think of that possibility
Director
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Posts: 789
Re: How many three digit numbers are divisible by 78 or 91?  [#permalink]

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12 May 2007, 18:50
Sergey_is_cool wrote:
Thanks saurabhmalpani i didn't think of that possibility

Thank you both, "Sergey_is_cool" for reducing my 1 wrong count and "Saurabhmalpani" for another.
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Re: How many three digit numbers are divisible by 78 or 91?  [#permalink]

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13 Mar 2016, 20:34
I did this way...
first thing - find least common multiple of both numbers:
78 - 2x39 = 2x3x13
91 - 7x13
so LCM is 2x3x7x13 = 78x7=70x7 + 8x7 = 490+56 = 546.

how many 3 digit multiples 91 has?
91x10 = 910, thus, we know that there are 9 multiples of 9 greater than 91, 3 digit.
how many 3 digit multiples 78 has?
78x10 = 780, so at least 9.
780+78 = 858 - 10
858+78 = 936 -11

now, there are 9+11-1 (546) = 19 numbers that are either divisible by 78 or by 91.
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Posts: 91
Re: How many three digit numbers are divisible by 78 or 91?  [#permalink]

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13 Mar 2016, 21:54
vshaunak@gmail.com wrote:
How many three digit numbers are divisible by 78 or 91?

A. 17
B. 19
C. 20
D. 21
E. 22

The least 3 digit number is 999.
There are 999/78 = 12 numbers which are divisible by 78
There are 999/91 = 10 numbers which are divisible by 91
The LCM of 78 and 91 is 546, which is divisible by both the numbers and that is the only 3 digit number.
We have eliminate two 2 digit numbers which are 78 and 91 in the list
That gives 12+10-1-2 = 19
Option B is correct
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Re: How many three digit numbers are divisible by 78 or 91?  [#permalink]

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14 Mar 2016, 00:47
Excellent Question...
Here We need to subtract one from the final answer as 546 which is divisible by both the numbers and is counted twice.
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Re: How many three digit numbers are divisible by 78 or 91?  [#permalink]

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03 Aug 2019, 03:02
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Re: How many three digit numbers are divisible by 78 or 91?   [#permalink] 03 Aug 2019, 03:02
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