Which of these must the factor of the product of four consecutive even

Question

Which of these must the factor of the product of four consecutive even integers:-
1) 48
2) 64
3) 96
4) 192
5) 80

A) 1,2 only
B) 2,3 only
C) 1,2,3 only
D) 1,2,3,4 only
E) All of them

abhimahna · Answer

Product of every 4 even consecutive multiples will have atleast 2^3 and 3 as its prime factor.

There 1,2,3 and 4 satisfy the condition.

1) 48 --> 2^4 * 3
2) 64 --> 2^6
3) 96 -->2^5 * 3
4) 192 -->2^6 * 3
5) 80 --> 2^4 * 5

5 may or may not hold true. Consider 4 multiples as 2,4,6 and 8. Here we don't have any 5 as a factor for the product of these multiples but for 80 we need one 5 as a factor, hence, FALSE.

ameyaprabhu · Answer

Let the four variables be a< b< c< d.

Assume a worst case scenario where the a equal to a prime number (hence odd).

therefore

a = divisible by 1
b = a + 1 (divisible by 2) 
c = a + 3 (divisible by 3)
d = a + 4 (divisible by 4)

therefore

each answer choice must be divisible by 2 x 3 x 4 = 24

only 80 is not divisible. Therefore answer = D

stonecold · Answer

Here is what id did in this Question ->
Let the 4 consecutive evens be ->
2n
2n+2
2n+4
2n+6

Taking their product => 2n*(2n+2)*(2n+4)*(2n+6)=> 2^4 n(n+1)(n+2)(n+3)

Property in action ->  Product of n consecutive integers is always divisible by n!

Hence => 2^4 n(n+1)(n+2)(n+3)=2^4*24k 
For some integer k.
Clearly Options 1,2,3,4 must be factors of 2^4*24k 
But 80 has a 5 in it so it may or may nor be the factor of 2^4*24k 
Hence we ned to exclude E from the answer choice.

Hence D.

gracie · Answer

say 2*4*6*8 
this gives us
6*8=48
2*4*8=64
2*6*8=96
4*6*8=192
but not 5, so not 80
1,2,3,4 only
D

abc568 · Answer

I think the question might be wrong here. Because, it dosnt mention, 'positive consecutive even numbers;
so, the lowest possible value can be 0,2,4,6; the product of which adds upto 48, & since the question is a 'must be', that means only 48 suffices as the correct answer.

Please let me know if I am missing something, because I am unsure.

Which of these must the factor of the product of four consecutive even

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Which of these must the factor of the product of four consecutive even

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards