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The sum of four natural numbers is 305. The ratios of these numbers in pairs are 2 : 3, 3 : 4, 4 : 5, 5 : 6, 8 : 9 and 15 : 16. What is smallest natural number that can be divided by these four numbers? (in numerical value)
Correct answer is '3600'. Can you explain this answer?
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The sum of four natural numbers is 305. The ratios of these numbers in...
Since one of the ratios in pairs is 15:16, we know that one of the numbers is a multiple of 15 and one other number is a multiple of 16.
The number which is a multiple of 15, when taken with the remaining two numbers should give a ratio of 4:5 and 5:6, which can be rewritten as 12:15 and 15:18 respectively. Thus, the ratio of the four numbers is 12 : 15 : 16 : 18.
Since the ratio adds up to 61 and the sum of the four numbers is 305 = 61 × 5, we have a common factor of 5.
The LCM of 12, 15, 16 and 18 is 720. Thus the smallest natural number that can be divided by the four natural numbers is 720 × 5 = 3600.
Alternately, we can determine that the four natural numbers are 60, 75, 80 and 90 and their LCM is 3600.
Most Upvoted Answer
The sum of four natural numbers is 305. The ratios of these numbers in...
Given Information:
- The sum of four natural numbers is 305.
- The ratios of these numbers in pairs are 2:3, 3:4, 4:5, 5:6, 8:9, and 15:16.
To Find:
The smallest natural number that can be divided by these four numbers.
Step-by-Step Solution:
Step 1:
Let's assume the four numbers as 2x, 3x, 4x, and 5x (where x is a common factor).
Step 2:
According to the given information, the sum of these four numbers is 305.
So, we have the equation:
2x + 3x + 4x + 5x = 305
Simplifying the equation, we get:
14x = 305
Step 3:
To find the value of x, divide both sides of the equation by 14:
x = 305/14
Step 4:
The value of x is not a whole number. Since we are required to find the smallest natural number that can be divided by these four numbers, we need to find the least common multiple (LCM) of the four numbers.
Step 5:
To find the LCM, we need to find the prime factors of each of the four numbers.
Prime factors of 2x: 2
Prime factors of 3x: 3
Prime factors of 4x: 2^2, 2
Prime factors of 5x: 5
Step 6:
To find the LCM, we take the highest power of each prime factor:
LCM = 2^2 * 3 * 5 = 60
Step 7:
To find the smallest natural number that can be divided by these four numbers, we multiply the LCM by x:
Smallest natural number = 60 * (305/14) = 60 * 21.79
Rounding up to the nearest whole number, we get:
Smallest natural number = 60 * 22 = 1320
However, the correct answer is given as 3600, which means we made an error in the calculations.
Correction:
Upon reevaluating the calculations, it appears that the LCM should be multiplied by the value of x before rounding up to the nearest whole number.
Smallest natural number = 60 * (305/14) = 60 * 21.79 ≈ 1311.4
Rounding up to the nearest whole number, we get:
Smallest natural number = 60 * 22 = 1320
Therefore, the correct answer is 1320, not 3600 as stated.
- The sum of four natural numbers is 305.
- The ratios of these numbers in pairs are 2:3, 3:4, 4:5, 5:6, 8:9, and 15:16.
To Find:
The smallest natural number that can be divided by these four numbers.
Step-by-Step Solution:
Step 1:
Let's assume the four numbers as 2x, 3x, 4x, and 5x (where x is a common factor).
Step 2:
According to the given information, the sum of these four numbers is 305.
So, we have the equation:
2x + 3x + 4x + 5x = 305
Simplifying the equation, we get:
14x = 305
Step 3:
To find the value of x, divide both sides of the equation by 14:
x = 305/14
Step 4:
The value of x is not a whole number. Since we are required to find the smallest natural number that can be divided by these four numbers, we need to find the least common multiple (LCM) of the four numbers.
Step 5:
To find the LCM, we need to find the prime factors of each of the four numbers.
Prime factors of 2x: 2
Prime factors of 3x: 3
Prime factors of 4x: 2^2, 2
Prime factors of 5x: 5
Step 6:
To find the LCM, we take the highest power of each prime factor:
LCM = 2^2 * 3 * 5 = 60
Step 7:
To find the smallest natural number that can be divided by these four numbers, we multiply the LCM by x:
Smallest natural number = 60 * (305/14) = 60 * 21.79
Rounding up to the nearest whole number, we get:
Smallest natural number = 60 * 22 = 1320
However, the correct answer is given as 3600, which means we made an error in the calculations.
Correction:
Upon reevaluating the calculations, it appears that the LCM should be multiplied by the value of x before rounding up to the nearest whole number.
Smallest natural number = 60 * (305/14) = 60 * 21.79 ≈ 1311.4
Rounding up to the nearest whole number, we get:
Smallest natural number = 60 * 22 = 1320
Therefore, the correct answer is 1320, not 3600 as stated.
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Community Answer
The sum of four natural numbers is 305. The ratios of these numbers in...
Since one of the ratios in pairs is 15:16, we know that one of the numbers is a multiple of 15 and one other number is a multiple of 16.
The number which is a multiple of 15, when taken with the remaining two numbers should give a ratio of 4:5 and 5:6, which can be rewritten as 12:15 and 15:18 respectively. Thus, the ratio of the four numbers is 12 : 15 : 16 : 18.
Since the ratio adds up to 61 and the sum of the four numbers is 305 = 61 × 5, we have a common factor of 5.
The LCM of 12, 15, 16 and 18 is 720. Thus the smallest natural number that can be divided by the four natural numbers is 720 × 5 = 3600.
Alternately, we can determine that the four natural numbers are 60, 75, 80 and 90 and their LCM is 3600.
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The sum of four natural numbers is 305. The ratios of these numbers in pairs are2 : 3, 3 : 4, 4 : 5, 5 : 6, 8 : 9 and 15 : 16. What is smallest natural number that can be divided by these four numbers? (in numerical value)Correct answer is '3600'. Can you explain this answer?
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The sum of four natural numbers is 305. The ratios of these numbers in pairs are2 : 3, 3 : 4, 4 : 5, 5 : 6, 8 : 9 and 15 : 16. What is smallest natural number that can be divided by these four numbers? (in numerical value)Correct answer is '3600'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The sum of four natural numbers is 305. The ratios of these numbers in pairs are2 : 3, 3 : 4, 4 : 5, 5 : 6, 8 : 9 and 15 : 16. What is smallest natural number that can be divided by these four numbers? (in numerical value)Correct answer is '3600'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of four natural numbers is 305. The ratios of these numbers in pairs are2 : 3, 3 : 4, 4 : 5, 5 : 6, 8 : 9 and 15 : 16. What is smallest natural number that can be divided by these four numbers? (in numerical value)Correct answer is '3600'. Can you explain this answer?.
The sum of four natural numbers is 305. The ratios of these numbers in pairs are2 : 3, 3 : 4, 4 : 5, 5 : 6, 8 : 9 and 15 : 16. What is smallest natural number that can be divided by these four numbers? (in numerical value)Correct answer is '3600'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The sum of four natural numbers is 305. The ratios of these numbers in pairs are2 : 3, 3 : 4, 4 : 5, 5 : 6, 8 : 9 and 15 : 16. What is smallest natural number that can be divided by these four numbers? (in numerical value)Correct answer is '3600'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of four natural numbers is 305. The ratios of these numbers in pairs are2 : 3, 3 : 4, 4 : 5, 5 : 6, 8 : 9 and 15 : 16. What is smallest natural number that can be divided by these four numbers? (in numerical value)Correct answer is '3600'. Can you explain this answer?.
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