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# If a and b are positive integers, is 4a + 3b divisible by 12?

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Math Expert
Joined: 02 Sep 2009
Posts: 60460
If a and b are positive integers, is 4a + 3b divisible by 12?  [#permalink]

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03 Dec 2019, 01:24
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Difficulty:

65% (hard)

Question Stats:

63% (02:34) correct 37% (01:55) wrong based on 45 sessions

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If a and b are positive integers, is $$4a + 3b$$ divisible by 12?

(1) $$b!$$ is a multiple of 3

(2) $$2a+5b=26$$

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Re: If a and b are positive integers, is 4a + 3b divisible by 12?  [#permalink]

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03 Dec 2019, 03:04
1
Bunuel wrote:
If a and b are positive integers, is $$4a + 3b$$ divisible by 12?

(1) $$b!$$ is a multiple of 3

(2) $$2a+5b=26$$

Are You Up For the Challenge: 700 Level Questions

#1
b! multiple of 3 ; b can be 3,4,5,6...
a not given insufficient
#2
2a+5b=26
b=2, a ; 13 ; b=4 a 3 ; we get yes and no
insufficient
from 1 &2
b has to be 4 ; so $$4a + 3b$$ is divisible by 12
sufficient
IMO C
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If a and b are positive integers, is 4a + 3b divisible by 12?  [#permalink]

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03 Dec 2019, 03:53
Constraint: a,b =+ve integer
Q: Is 4(a)+3(b)/12 =Integer?

(1) 3! =6 , 4! =24, 5!=120...(mult.of 3)
.: b= 3,4,5 .... but a = any +ve integer
(Not sufficient)

(2) 2a =26-5b —> a= (26-5b)/2= +ve Integer ,so b< 5
a =(26-5(4))/2 =3. Here a=3,b=4
Divisible by 12
Or
a= (26-5(2)/2=16/2=8. Here a=8 ,b=2. Not divisible by 12
(Not Sufficient)

(1+2) a= (26-5(4))/2=6/2 =3
Since subt. 3 and 5 will make a non-Integer
.: a=3 ,b=4
4(3)+3(4)/12 = Integer

Bunuel wrote:
If a and b are positive integers, is $$4a + 3b$$ divisible by 12?

(1) $$b!$$ is a multiple of 3

(2) $$2a+5b=26$$

Are You Up For the Challenge: 700 Level Questions

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Joined: 24 Nov 2016
Posts: 1066
Location: United States
Re: If a and b are positive integers, is 4a + 3b divisible by 12?  [#permalink]

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04 Dec 2019, 09:05
Bunuel wrote:
If a and b are positive integers, is $$4a + 3b$$ divisible by 12?

(1) $$b!$$ is a multiple of 3

(2) $$2a+5b=26$$

4a+3b and must be an even to be divisible by 12
4a=even, so 3b=even, and b=even

(1) $$b!$$ is a multiple of 3 insufic

b!=m(3): b=even≥3:{4,6,8,…}

(2) $$2a+5b=26$$ insufic

2a+5b=26, 2a=26-5b, a=13-5b/2>0 (b=even)
b=2: a=13-5=8; 4a+3b=32+6≠m(12)
b=4: a=13-10=3; 4a+3b=12+12=m(12)

(1)&(2) sufic

b=even≥3:{4,6,8,…}=4
b=4: a=13-10=3; 4a+3b=12+12=m(12)

Ans (C)
Re: If a and b are positive integers, is 4a + 3b divisible by 12?   [#permalink] 04 Dec 2019, 09:05
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