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If a, b, c, d, and e are integers and the expression (ab2c2 /d2e) gives a positive even integer, which of the following options must be true?
I. abc is even
II. a/e is positive
III. a/d2 is positive
- a)I only
- b)II only
- c)I and II
- d)II and III
- e)I, II, and III
Correct answer is 'C'. Can you explain this answer?
Verified Answer
If a, b, c, d, and e are integers and the expression (ab2c2 /d2e) give...
Step 1: Question statement and Inferences
We are given that a, b, c, d, and e are integers.
Also, it is given that
ab2c2 /d2e = positive even number . . .. (1)
We can also write this expression as:
ab2c2 = (Positive Even number)*(d2e)
When you multiply any number, whether it is even or odd, with an even number, the product is always even.
So, we can deduce that:
- ab2c2 = even
Also, equation (1) can be written as:
(b2c2 /d2) * (a/e) = positive even number
Now, we know that the product of two numbers can be positive only if both are positive or both are negative: (Think; 2*3 = 6 OR -2 * -3 = 6)
And in the above expression, (b2c2 /d2) is always positive since even powers of an integer are always positive.
Since the first number (b2c2 /d2) is always positive, the second number (a/e) also needs to be positive.
(Note that, here it is possible that a and e both are negative, but the overall expression (a/e) is definitely positive)
So, we know that a/e is positive.
Step 2: Finding required values
Now, let’s move on to the analysis of the options:
I. abc is even.
As we know, the expression (ab2c2) is even and it can be written as:
ab2c2 = a*b*b*c*c
Now, since the above expression yields an even number, it tells us that at least one of the integers a, b, or c is an even number. (Think 2 * 3 * 5 = 30)
Now, if at least one of the integers a, b, or c is an even number then the product (a*b*c) will also be an even number.
So, this statement is always true.
II. a/e is positive
We already proved that a/e is positive.
So, this statement is always true. III. a/d2 is positive
In the expression a/d2 since d2 is a square form, it is always positive. However, the question doesn’t state whether a is positive or negative. So, we can’t clearly say whether a/d2 is positive or negative.
So, we can’t say that this statement is always true.
Step 3: Calculating the final answer
Hence, the options that are always true are I and II. So, the correct answer choice is choice C.
Answer: Option (C)
Take Away
- When a fraction after simplification can be represented as an even integer, it implies that the numerator of that fraction is an even integer.
- If a multiplication expression yields a positive product, it means that the number of negative numbers in the expression is even. It can be zero, two, four etc. but it will never be an odd number. For example:
(-2) * (-1) * (3) = 6
(-3) * (4) * (5) = -60
Most Upvoted Answer
If a, b, c, d, and e are integers and the expression (ab2c2 /d2e) give...
If the expression (ab^2c^2 / d^2e) gives a positive even integer, we can analyze the factors in the expression to determine which options must be true.
I. abc is even:
- For the expression (ab^2c^2 / d^2e) to be even, there must be at least one 2 in the numerator (ab^2c^2) or denominator (d^2e). This means that either a, b, or c must be even.
- If a, b, or c is even, then abc will be even. Therefore, option I must be true.
II. a/e is positive:
- To determine whether a/e is positive, we need to consider the signs of a and e.
- Since the expression (ab^2c^2 / d^2e) gives a positive even integer, both the numerator and denominator must have the same sign (either both positive or both negative).
- If both a and e have the same sign, then a/e will be positive. Therefore, option II must be true.
III. a/d^2 is positive:
- Similar to the analysis for option II, we need to consider the signs of a and d.
- Since the expression (ab^2c^2 / d^2e) gives a positive even integer, both the numerator and denominator must have the same sign.
- If both a and d have the same sign, then a/d^2 will be positive. Therefore, option III must be true.
Therefore, options I and II must be true, and option III must also be true. Hence, the correct answer is option C (I and II).
I. abc is even:
- For the expression (ab^2c^2 / d^2e) to be even, there must be at least one 2 in the numerator (ab^2c^2) or denominator (d^2e). This means that either a, b, or c must be even.
- If a, b, or c is even, then abc will be even. Therefore, option I must be true.
II. a/e is positive:
- To determine whether a/e is positive, we need to consider the signs of a and e.
- Since the expression (ab^2c^2 / d^2e) gives a positive even integer, both the numerator and denominator must have the same sign (either both positive or both negative).
- If both a and e have the same sign, then a/e will be positive. Therefore, option II must be true.
III. a/d^2 is positive:
- Similar to the analysis for option II, we need to consider the signs of a and d.
- Since the expression (ab^2c^2 / d^2e) gives a positive even integer, both the numerator and denominator must have the same sign.
- If both a and d have the same sign, then a/d^2 will be positive. Therefore, option III must be true.
Therefore, options I and II must be true, and option III must also be true. Hence, the correct answer is option C (I and II).
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