If a and b are even integers, which of the following is an odd integer

Answer = E. (a + 3)(b − 1)

Say a = 2, b = 2

A >> 2*2 + 2 = 6 >> Even

B & C >> No need to check; it getting multiplied directly by 2, so would always be even

D >> 3*2 + 4*2 = 6+8 = 14 >> Even

E >> (2+3)(2-1) = 5 * 1 = 5 >> ODD

Testing all the option individually :
Given both a & b are even
Let a=2 & b=4

A. ab + 2
EVEN
B. a(b − 1)
Product of even and odd is EVEN.
C. a(a + 5)
Product of even and odd is EVEN
D. 3a + 4b
EVEN
E. (a + 3)(b − 1)
Product of odd and odd is ODD.

Answer is E

a and b are even integers.

A) ab + 2 => even*even+ even => even+even => even
B) a(b − 1) => even(even-1) => even*odd => even
C) a(a + 5) => even(even+odd) => even
D) 3a + 4b => odd*even+even*even => even
E) (a + 3)(b − 1) => odd*odd => odd

Answer E

Official Solution:

If a and b are even integers, which of the following is an odd integer?

A) ab + 2
B) a(b − 1)
C) a(a + 5)
D) 3a + 4b
E) (a + 3)(b − 1)

Picking Numbers

This key strategy involves first picking numbers and then substituting them into the answer choices. Whenever a problem involves variables, we may consider using this strategy. For this particular problem, pick the numbers a = 2 and b = 4 because both are even integers, yet both are still small and manageable numbers. Now substitute. Answer choice E is correct. You can be confident that if it works for your chosen set of numbers, it will also work for all other numbers as well. There is no need to try other numbers.

ab + 2 (2 × 4) + 2 = 10 even
a(b − 1) 2(4 − 1) = 6 even
a(a + 5) 2(2 + 5) = 14 even
3a + 4b (3 × 2) + (4 × 4) = 22 even
(a + 3)(b − 1) (2 + 3)(4 − 1) = 15 odd

Answer: E.

Option E

a,b are Even integers. Odd: O, Even: E & Fraction: F

A. ab + 2 = E*E + 2 = E + 2 = E
B. a(b − 1) = E*(E - 1) = E
C. a(a + 5) = E*(E + 5) = E
D. 3a + 4b = 3*E + 4*E = E
E. (a + 3)(b − 1) = (E +3)*(E - 1) = O*O = O

If a and b are even integers, which of the following is an odd integer?

E. (a + 3)(b − 1)
(even + 3)(even - 1)
odd *odd = odd

EVEN * EVEN = EVEN
EVEN * ODD = EVEN
EVEN + EVEN = EVEN
EVEN + ODD = ODD
ODD * ODD = ODD

'a' and 'b' are even:

Option A: Even * Even + Even = Even

Option B: Even ( Even -1): Even * Odd = Even

Option C: Even( Even + Odd): Even * Odd = Even

Option D: 3*Even + 4 Even: Even + Even = Even

Option E: (Even + Odd)(Even - Odd): Odd * Odd = ODD

Answer E

Solution:

We recall 5 arithmetic facts: (1) even +/- even = even (2) even +/- odd = odd (3) even x even = even (4) even x odd = even (5) odd x odd = odd

Let’s evaluate each answer choice, using these arithmetic facts.

Choice A: ab + 2 . Using (3) and (1), we see that this answer is even.

Choice B: a(b - 1). Using (2) and (4), we see that this answer is even.

Choice C: a(a + 5) . Using (2) and (4), we see that this answer is even.

Choice D: 3a + 4b . Using (4), (3), and (1), we see that this answer is even.

Choice E: (a + 3)(b - 1). Using (2), (2), and (5), we see that this answer is odd.

Answer: E