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If both the product and sum of four integers are even, which of the following could be the number of even integers in the group?

I. 0 II. 2 III. 4

A. I only B. II only C. III only D. II and III only E. I, II, and III

hi, i)for the product of any number of integers to be even, only one number requires to be even... ii) for any set numbers to have their sum to be even, the number ofodd numbers should be even...

with these two properties in mind.. for four int to have their product and sum as even.. 2 and 4 are ok.. but 0 even number will result in an odd integer as product.. D
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Re: If both the product and sum of four integers are even, which of the [#permalink]

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15 Jan 2017, 23:34

Nice Question. Here is what i did in this one => Since the product is even => At-least one of them must be even. Since the sum is even => Number of odd numbers must be even

If both the product and sum of four integers are even, which of the following could be the number of even integers in the group?

I. 0 II. 2 III. 4

A. I only B. II only C. III only D. II and III only E. I, II, and III

Let’s test each Roman numeral.

I. 0

Since we need at least 1 even integer in the set for the product of 4 integers to be even, there could not be 0 even integers in the set.

II. 2

If there are 2 even numbers in the set, there also would be 2 odd numbers. Furthermore, the sum of 2 even integers and 2 odd integers is even, and the product of 2 even integers and 2 odd integers is even. There could be 2 even integers in the set.

III. 4

Since the sum of 4 even integers is even and the product of 4 even integers is even, there could be 4 even integers in the set.

Answer: D
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If both the product and sum of four integers are even, which of the following could be the number of even integers in the group?

I. 0 II. 2 III. 4

A. I only B. II only C. III only D. II and III only E. I, II, and III

Given: a + b + c + d = even and a*b*c*d = even.

For the sum of 4 integers to be even, group should contain 0, 2 or 4 even numbers. So possible scenarios are 0, 2, or 4 even numbers among 4.

For the product of the integers to be even at least one of them should be even. So 1, 2, 3, or all 4 numbers from a,b,c,d should be even. If there is 0 even number among them, it means that all 4 integers are odd, the product of four odd integers is odd. Hence there can not be 0 even number. So possible scenarios are 1, 2, 3 or 4 even numbers among 4.

Both conditions to be met: there can be 2 or 4 even numbers among 4.