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How many natural numbers less than 10 when squared and then divided by 24 leaves a remainder of 1?
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How many natural numbers less than 10 when squared and then divided by...
Understanding the Problem
We need to find the natural numbers less than 10 whose squares, when divided by 24, leave a remainder of 1. In other words, we are looking for numbers \( n \) such that:
n^2 mod 24 = 1
Natural Numbers to Consider
The natural numbers less than 10 are:
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9
Calculating Squares and Remainders
Now, let’s calculate the square of each number and find the remainder when divided by 24:
- 1^2 = 1; 1 mod 24 = 1
- 2^2 = 4; 4 mod 24 = 4
- 3^2 = 9; 9 mod 24 = 9
- 4^2 = 16; 16 mod 24 = 16
- 5^2 = 25; 25 mod 24 = 1
- 6^2 = 36; 36 mod 24 = 12
- 7^2 = 49; 49 mod 24 = 1
- 8^2 = 64; 64 mod 24 = 16
- 9^2 = 81; 81 mod 24 = 9
Numbers that Meet the Condition
From the calculations above, the numbers that satisfy the condition n^2 mod 24 = 1 are:
- 1 (1 mod 24 = 1)
- 5 (25 mod 24 = 1)
- 7 (49 mod 24 = 1)
Total Count
Thus, there are three natural numbers less than 10 that, when squared and divided by 24, leave a remainder of 1:
- 1
- 5
- 7
Conclusion
The final answer is 3.
We need to find the natural numbers less than 10 whose squares, when divided by 24, leave a remainder of 1. In other words, we are looking for numbers \( n \) such that:
n^2 mod 24 = 1
Natural Numbers to Consider
The natural numbers less than 10 are:
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9
Calculating Squares and Remainders
Now, let’s calculate the square of each number and find the remainder when divided by 24:
- 1^2 = 1; 1 mod 24 = 1
- 2^2 = 4; 4 mod 24 = 4
- 3^2 = 9; 9 mod 24 = 9
- 4^2 = 16; 16 mod 24 = 16
- 5^2 = 25; 25 mod 24 = 1
- 6^2 = 36; 36 mod 24 = 12
- 7^2 = 49; 49 mod 24 = 1
- 8^2 = 64; 64 mod 24 = 16
- 9^2 = 81; 81 mod 24 = 9
Numbers that Meet the Condition
From the calculations above, the numbers that satisfy the condition n^2 mod 24 = 1 are:
- 1 (1 mod 24 = 1)
- 5 (25 mod 24 = 1)
- 7 (49 mod 24 = 1)
Total Count
Thus, there are three natural numbers less than 10 that, when squared and divided by 24, leave a remainder of 1:
- 1
- 5
- 7
Conclusion
The final answer is 3.
Community Answer
How many natural numbers less than 10 when squared and then divided by...
3
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How many natural numbers less than 10 when squared and then divided by 24 leaves a remainder of 1?
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How many natural numbers less than 10 when squared and then divided by 24 leaves a remainder of 1? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about How many natural numbers less than 10 when squared and then divided by 24 leaves a remainder of 1? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many natural numbers less than 10 when squared and then divided by 24 leaves a remainder of 1?.
How many natural numbers less than 10 when squared and then divided by 24 leaves a remainder of 1? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about How many natural numbers less than 10 when squared and then divided by 24 leaves a remainder of 1? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many natural numbers less than 10 when squared and then divided by 24 leaves a remainder of 1?.
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