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# A 6-digit number comprises of only 2’s and 3’s. How many of

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VP
Joined: 20 Jul 2017
Posts: 1096
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
A 6-digit number comprises of only 2’s and 3’s. How many of  [#permalink]

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23 Oct 2019, 03:22
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Difficulty:

65% (hard)

Question Stats:

44% (02:14) correct 56% (02:11) wrong based on 55 sessions

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A 6-digit number comprises of only 2’s and 3’s. How many of these are multiples of 12?

A. 0
B. 1
C. 6
D. 20
E. 21

Posted from my mobile device
Math Expert
Joined: 02 Aug 2009
Posts: 8202
Re: A 6-digit number comprises of only 2’s and 3’s. How many of  [#permalink]

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23 Oct 2019, 03:39
2
3
Dillesh4096 wrote:
A 6-digit number comprises of only 2’s and 3’s. How many of these are multiples of 12?

A. 0
B. 1
C. 6
D. 20
E. 21

Posted from my mobile device

For it to be a multiple of 12, the number has to be multiple of 3 and 4..
1) Multiple of 4...
Last two digits have to be multiple of 4....Only combination is 32....ABCD32 .. 1 way
2) Multiple of 3..
Since we have used one 2 in 32, we will require to use 2 more 2s in first four digits..
So, ABCD can take two 2s and two 3s in 4!/2!2!=6way

Total 1*6=6

C
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VP
Joined: 19 Oct 2018
Posts: 1083
Location: India
Re: A 6-digit number comprises of only 2’s and 3’s. How many of  [#permalink]

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23 Oct 2019, 03:40
1
Number must be a multiple of 4; hence, last 2 digits must me 32

Number must be a multiple of 3; hence there can be zero, three or six 2's in the number.

Case 1- numbers comprise of zero 2's and six 3's
Such number can't be a multiple of 4

Case 2- numbers comprise of three 2's and three 3's.
Last 2 digits are 32. We can arrange remaining two 2's and two 3's.

Total possible cases= $$\frac{4!}{2!2!}$$=6

Case 3- numbers comprise of six 2's.
such number can't be a multiple of 4

Total possible cases= 0+6+0=6

Dillesh4096 wrote:
A 6-digit number comprises of only 2’s and 3’s. How many of these are multiples of 12?

A. 0
B. 1
C. 6
D. 20
E. 21

Posted from my mobile device
Senior Manager
Joined: 21 Jun 2017
Posts: 307
Location: India
Concentration: Finance, Economics
Schools: IIM
GPA: 3
WE: Corporate Finance (Commercial Banking)
Re: A 6-digit number comprises of only 2’s and 3’s. How many of  [#permalink]

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23 Oct 2019, 08:43
1
1
Last two digits should be 32. Rest 4 digits should be 2233 form ----> 4!/(2!*2!) = 6 = answer
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Posts: 29
Re: A 6-digit number comprises of only 2’s and 3’s. How many of  [#permalink]

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25 Oct 2019, 12:10
Hello,

I have a silly question, how do you guys know that the 4 first number must include two 2's and two 3's?

Math Expert
Joined: 02 Aug 2009
Posts: 8202
Re: A 6-digit number comprises of only 2’s and 3’s. How many of  [#permalink]

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25 Oct 2019, 19:17
azl wrote:
Hello,

I have a silly question, how do you guys know that the 4 first number must include two 2's and two 3's?

Hi,

We have 32 A’s last two digits.
The first 4 can be any of 3 and 2, but all 6 when added should give you a multiple of 3, a property for multiple of 3.
Now you can have as many 3s as they are multiple of 3, but when you have a 2, either you have a 1 with it as 2+1=3 but that is not possible. So you have to have three or six etc of 2. Here you can have three 2s and one 2 is already used in units digit abcd32. So ABCD can have two 2s and the remaining will be 3s

Posted from my mobile device
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Re: A 6-digit number comprises of only 2’s and 3’s. How many of   [#permalink] 25 Oct 2019, 19:17
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