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If a, b, and c are consecutive integers and 0 < a < b < c, is the product abc a multiple of 8 ?

(1) The product ac is even.

(2) The product bc is a multiple of 4.

hi,

If a, b, and c are consecutive integers and 0 < a < b < c, is the product abc a multiple of 8 TWO cases.. A) a and c are even AND b is ODD.... abc will always be multiple of 8 or rather 24.. ans will be YES B) a and c are ODD and b is even.. if b is multiple of 8.. YES otherwise NO

lets see the statements (1) The product ac is even. this MEANS a and c are even.. falls under category A above.. always YES sufficient (2) The product bc is a multiple of 4 either b is multiple of 4, ans can be YES or NO case B OR c is multiple of 4, ans will be YES.. case A different answers possible insuff

as the numbers are consecutive, so \(a\), \(b\), \(c\) can take two options -

Case 1: Even, Odd, Even OR

Case 2: Odd, Even, Odd

Statement 1: as \(ac\) is even, so it has to be Case 1

Now let \(a=2k\) (even number) so \(b=2k+1\) and \(c=2k+2=2(k+1)\), where \(k\) is any number greater than \(0\) (because \(0<a<b<c\))

Hence \(ac=2k*2(k+1)=4k(k+1)\)

so if \(k\) is odd then \(k+1=even\), hence \(4k(k+1)\) will be a multiple of \(8\)

if \(k\) is even then \(4k\) will be a multiple of \(8\). Therefore \(abc\) is a multiple of \(8\) because \(ac\) is a multiple of \(8\). Sufficient

Statement 2: \(bc\) is a multiple of \(4\). so it could be multiple of \(8\) such as \(16,24,32\) etc. or it could not be a multiple of \(8\) such as \(20, 28\) etc.. Insufficient

Re: If a, b, and c are consecutive integers and 0 < a < b < c, is the prod [#permalink]

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01 Dec 2017, 20:29

carcass wrote:

If a, b, and c are consecutive integers and 0 < a < b < c, is the product abc a multiple of 8 ?

(1) The product ac is even.

(2) The product bc is a multiple of 4.

The Best would be to plug in by numbers.

Let b = n, a = n-1, c = n+1

Stmnt 1: ac = even, \(n^2\) - 1 =even, Here n can be only a odd number starting from 3,5,7,9,11 etc.. So abc will be Divisible by 8. Stmnt 2: n(n+1) = 4m.. Here, n can be 3,4,7,8,9,11,13,14.. etc. when n = 3, abc Divisible by 8 but n= 4, abc not divisible by 8.

I don't get why a and c are even AND b is ODD.... abc will always be multiple of 8 or rather 24, could you explain me please

Hi... Three consecutive integers and two even means even, odd, even... Now since three are consecutive, one of them would surely be a MULTIPLE of 3. Two of them even means one will be MULTIPLE of 2 and other will be MULTIPLE of atleast 4 so product will be MULTIPLE of atleast 3*2*4=24
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