Analyzing the Condition: a - b is Even
The condition that the difference between two integers, $a - b$, is even implies that $a$ and $b$ must have the same parity. This means either both $a$ and $b$ are even, or both $a$ and $b$ are odd.
Evaluating the Options
We need to find the expression that is always even, given that $a$ and $b$ have the same parity.
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Option 1: $ab$
- If $a$ and $b$ are both odd: $a * b$ = odd * odd = odd.
- Since this case results in an odd number, $ab$ is not always even.
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Option 2: $a2 + b2 + 1$
- If $a$ and $b$ are both even: $a2$ is even, $b2$ is even. So, $even + even + 1$ = odd.
- If $a$ and $b$ are both odd: $a2$ is odd, $b2$ is odd. So, $odd + odd + 1$ = $even + 1$ = odd.
- This expression is always odd.
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Option 3: $a2 + b + 1$
- If $a$ and $b$ are both even: $a2$ is even. So, $even + even + 1$ = odd.
- If $a$ and $b$ are both odd: $a2$ is odd. So, $odd + odd + 1$ = $even + 1$ = odd.
- This expression is always odd.
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Option 4: $ab - b$
- We can factor this expression as $b(a - 1)$.
- Case 1: Both $a$ and $b$ are even. Let $a = 2k$ and $b = 2m$ for some integers $k, m$. $ab - b$ = $(2k)(2m) - 2m$ = $4km - 2m$ = $2(2km - m)$. This is always even.
- Case 2: Both $a$ and $b$ are odd. Let $a = 2k + 1$ and $b = 2m + 1$ for some integers $k, m$. $a - 1$ = $(2k + 1) - 1$ = $2k$ (which is even). $ab - b$ = $b(a - 1)$ = $odd * even$ = even.
- Since the expression is even in both possible cases (both even or both odd), $ab - b$ must always be even.
Conclusion
Based on the analysis of parity, the expression $ab - b$ is the only one guaranteed to be even when $a - b$ is even.
Important Questions from Numerical Reasoning
- Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
- A 'frabjous' number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?
Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:
1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.
Let $F(N)$ denote the number of possible ways in which Ankita can reach the $N^{th}$ stair. For example, $F(1) = 1$, $F(2) = 2$, $F(3) = 3$. The value of $F(5)$ is ________.- In a zoo, three lions and four tigers eat 390 kg of food every week. In another zoo, four lions and five tigers eat 500 kg of food every week. Lions and tigers eat different amounts of food, but all individuals of the same species eat the same amount. The amount of food a single lion eats per week is ________ kg.
(Answer in integer) Consider a spherical globe rotating about an axis passing through its poles. There are three points P, Q, and R situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let P, Q, and R move with speeds $v_P$, $v_Q$, and $v_R$, respectively.
Which one of the following options is CORRECT?
