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For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.
If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit of 
- a)0
- b)1
- c)27
- d)54
- e)Cannot be determined
Correct answer is option 'E'. Can you explain this answer?
Verified Answer
For any positive number x, the function [x] denotes the greatest integ...
Given:
- Positive integer k (so, k > 0)
- k2 is divisible by 45
- k2 is divisible by 80
- The function [x]
To find: The units digit of 
Approach:
- To find the units digit of [x], we need to know or at least have some idea of the value of x. So, in order to answer the question, we need to find the value or at least a clue about the value of the number
- This number contains k3, which is the cube of a positive integer. However, the given information is about k2. Like in all questions that involve divisibility information about different powers of an integer, we’ll work out information about k3 as under:
- First use the information given about k2 to infer about the prime-factorized form of k.
- Then, by cubing the prime-factorized expression of k, get an expression for k3
Working Out:
- Inferring the prime-factorized expression for k from the given information
- Let k = P1a * P2b *P3c *P4d . . .where P1, P2 etc. are prime numbers and a, b . . . are non-negative integers
- Therefore, k2 = P12a * P22b *P32c *P42d . . .
- We are given that k2 is divisible by 45
- 45 = 32*5
- This means, P1 = 3 and 2a ≥ 2
- That is, a ≥ 1
- And, P2 = 5 and 2b ≥ 1
- That is,
- Since b is an integer, minimum possible value of b = 1
- That is,
- Therefore, k = (31*51)(3x*5y* P3c *P4d . . .) where x and y are non-negative integers
- Note: In the above expression, we’ve taken 31 and 51 outside the remaining expression in order to emphasize the fact that k definitely does contain 31 and 51, whether or not it contains higher powers of 3 and 5
- So far, we’ve inferred that: k = (31*51)(3x*5y* P3c *P4d . . .)
- So, k2 = (32*52)(32x*52y* P32c *P42d . . .)
- We are also given that k2 is divisible by 80
- 80 = 24*5
- This means, P3 = 2 and 2c ≥ 4
- That is, c ≥ 2
- So, k = (22*31*51)(2z*3x*5y* P4d . . .), where x, y, z, d etc. are non-negative integers
- Inferring the expression for k3
- Using the expression for k inferred above, we can write:
- k3 = (26*33*53)(23z*33x*53y* P43d . . .),
- Drawing inferences about the value of
- 4000 = 25*53
- Looking at the expression for k3 above, we see that k3 will be divisible by 4000
- So,
is an integer - The value of this integer will be (21*33)(23z*33x*53y* P43d . . .)
- So, = (21*33)(23z*33x*53y* P43d . . .)
- If y ≥ 1, then the units digit of the right hand side expression will be 0 (because the units digit of 2*5 is 0. Note that the above expression has at least one 2)
- But if y = 0, then the units digit will depend on the power of 2, 3 and the value and powers of any other prime numbers that are present in k
-
- Thus, we are not able to determine a unique value of the units digit of
- Looking at the answer choices, we see that the correct answer is Option E
- Let k = P1a * P2b *P3c *P4d . . .where P1, P2 etc. are prime numbers and a, b . . . are non-negative integers
Most Upvoted Answer
For any positive number x, the function [x] denotes the greatest integ...
Given:
- Positive integer k (so, k > 0)
- k2 is divisible by 45
- k2 is divisible by 80
- The function [x]
To find: The units digit of
Approach:
- To find the units digit of [x], we need to know or at least have some idea of the value of x. So, in order to answer the question, we need to find the value or at least a clue about the value of the number
- This number contains k3, which is the cube of a positive integer. However, the given information is about k2. Like in all questions that involve divisibility information about different powers of an integer, we’ll work out information about k3 as under:
- First use the information given about k2 to infer about the prime-factorized form of k.
- Then, by cubing the prime-factorized expression of k, get an expression for k3
Working Out:
- Inferring the prime-factorized expression for k from the given information
- Let k = P1a * P2b *P3c *P4d . . .where P1, P2 etc. are prime numbers and a, b . . . are non-negative integers
- Therefore, k2 = P12a * P22b *P32c *P42d . . .
- We are given that k2 is divisible by 45
- 45 = 32*5
- This means, P1 = 3 and 2a ≥ 2
- That is, a ≥ 1
- And, P2 = 5 and 2b ≥ 1
- That is,
- Since b is an integer, minimum possible value of b = 1
- That is,
- Therefore, k = (31*51)(3x*5y* P3c *P4d . . .) where x and y are non-negative integers
- Note: In the above expression, we’ve taken 31 and 51 outside the remaining expression in order to emphasize the fact that k definitely does contain 31 and 51, whether or not it contains higher powers of 3 and 5
- So far, we’ve inferred that: k = (31*51)(3x*5y* P3c *P4d . . .)
- So, k2 = (32*52)(32x*52y* P32c *P42d . . .)
- We are also given that k2 is divisible by 80
- 80 = 24*5
- This means, P3 = 2 and 2c ≥ 4
- That is, c ≥ 2
- So, k = (22*31*51)(2z*3x*5y* P4d . . .), where x, y, z, d etc. are non-negative integers
- Inferring the expression for k3
- Using the expression for k inferred above, we can write:
- k3 = (26*33*53)(23z*33x*53y* P43d . . .),
- Drawing inferences about the value of
- 4000 = 25*53
- Looking at the expression for k3 above, we see that k3 will be divisible by 4000
- So, is an integer
- The value of this integer will be (21*33)(23z*33x*53y* P43d . . .)
- So, = (21*33)(23z*33x*53y* P43d . . .)
- If y ≥ 1, then the units digit of the right hand side expression will be 0 (because the units digit of 2*5 is 0. Note that the above expression has at least one 2)
- But if y = 0, then the units digit will depend on the power of 2, 3 and the value and powers of any other prime numbers that are present in k
-
- Thus, we are not able to determine a unique value of the units digit of
- Looking at the answer choices, we see that the correct answer is Option E
- Let k = P1a * P2b *P3c *P4d . . .where P1, P2 etc. are prime numbers and a, b . . . are non-negative integers
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Question Description
For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer?.
For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer?.
Solutions for For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer? in English & in Hindi are available as part of our courses for UPSC.
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Here you can find the meaning of For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer?, a detailed solution for For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer? has been provided alongside types of For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.If k is a positive integer such that k2 is divisible by 45 and 80, what is the units digit ofa)0b)1c)27d)54e)Cannot be determinedCorrect answer is option 'E'. Can you explain this answer? tests, examples and also practice UPSC tests.
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