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If k and y are integers, and 10k + y is odd, which
of the following must be true?
a)k is odd
b)k is even
c)y is odd
d)both k and y are odd
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If k and y are integers, and 10k + y is odd, which... more of the foll...
Because Y is unit digit and we know that the unit digit of an Odd no is always odd.
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If k and y are integers, and 10k + y is odd, which... more of the foll...
Explanation:
To determine which of the given statements must be true, let's analyze the information provided and the possible scenarios:
Given:
- k and y are integers
- 10k + y is odd
Scenario 1: k is odd
- If k is odd, it can be represented as k = 2n + 1, where n is an integer.
- Substituting k in the expression 10k + y, we get 10(2n + 1) + y = 20n + 10 + y = 2(10n + 5) + y = 2m + y, where m = 10n + 5 is an integer.
- Since 2m is even, the sum 2m + y can only be odd if y is odd.
- Therefore, if k is odd, y must be odd.
Scenario 2: k is even
- If k is even, it can be represented as k = 2n, where n is an integer.
- Substituting k in the expression 10k + y, we get 10(2n) + y = 20n + y = 2(10n) + y = 2m + y, where m = 10n is an integer.
- In this case, the sum 2m + y can be odd or even depending on the value of y.
- Therefore, if k is even, y can be either odd or even.
Conclusion:
From the analysis of the two scenarios, we can conclude that the only statement that must be true is:
c) y is odd
This is because if 10k + y is odd, y must be odd in order to make the sum odd, regardless of whether k is odd or even.
Therefore, option 'C' is the correct answer.
To determine which of the given statements must be true, let's analyze the information provided and the possible scenarios:
Given:
- k and y are integers
- 10k + y is odd
Scenario 1: k is odd
- If k is odd, it can be represented as k = 2n + 1, where n is an integer.
- Substituting k in the expression 10k + y, we get 10(2n + 1) + y = 20n + 10 + y = 2(10n + 5) + y = 2m + y, where m = 10n + 5 is an integer.
- Since 2m is even, the sum 2m + y can only be odd if y is odd.
- Therefore, if k is odd, y must be odd.
Scenario 2: k is even
- If k is even, it can be represented as k = 2n, where n is an integer.
- Substituting k in the expression 10k + y, we get 10(2n) + y = 20n + y = 2(10n) + y = 2m + y, where m = 10n is an integer.
- In this case, the sum 2m + y can be odd or even depending on the value of y.
- Therefore, if k is even, y can be either odd or even.
Conclusion:
From the analysis of the two scenarios, we can conclude that the only statement that must be true is:
c) y is odd
This is because if 10k + y is odd, y must be odd in order to make the sum odd, regardless of whether k is odd or even.
Therefore, option 'C' is the correct answer.
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