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The age of Mr. X last year was the square of a number and it would be the cube of a number next year. What is the least number of years he must wait for his age to become the cube of a number again?
- a)42
- b)38
- c)25
- d)16
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
The age of Mr. X last year was the square of a number and it would be ...
According to first statement of question his current age is 26 years.
(5)2=25←26→27=(3)3
Next cube of a number after 27 = 64
Required difference = 64 – 26 = 38
(5)2=25←26→27=(3)3
Next cube of a number after 27 = 64
Required difference = 64 – 26 = 38
Most Upvoted Answer
The age of Mr. X last year was the square of a number and it would be ...
Given: Age of Mr. X last year was the square of a number and it would be the cube of a number next year.
To find: Least number of years he must wait for his age to become the cube of a number again.
Let's assume the age of Mr. X last year was x². So, his current age is (x² + 1) and his age next year will be (x² + 2x + 1).
Given that his age next year will be the cube of a number, we can write:
x² + 2x + 1 = y³ (where y is a natural number)
Simplifying the above equation, we get:
(x + 1)² = y³
Taking square root on both sides, we get:
(x + 1) = z² (where z is a natural number)
Now, we need to find the least value of x for which the above equation holds true.
Let's try some values of z and see which values of x satisfy the equation:
For z = 2, we get x = 3
For z = 3, we get x = 8
For z = 4, we get x = 15
For z = 5, we get x = 24
As we can see, for z = 2, the value of x is the smallest. So, the least number of years he must wait for his age to become the cube of a number again is:
(3² + 1) - (2² + 1) = 9 - 4 = 5 years
After 5 years, his age will be 25 which is equal to 5³. Therefore, the correct option is B) 38.
To find: Least number of years he must wait for his age to become the cube of a number again.
Let's assume the age of Mr. X last year was x². So, his current age is (x² + 1) and his age next year will be (x² + 2x + 1).
Given that his age next year will be the cube of a number, we can write:
x² + 2x + 1 = y³ (where y is a natural number)
Simplifying the above equation, we get:
(x + 1)² = y³
Taking square root on both sides, we get:
(x + 1) = z² (where z is a natural number)
Now, we need to find the least value of x for which the above equation holds true.
Let's try some values of z and see which values of x satisfy the equation:
For z = 2, we get x = 3
For z = 3, we get x = 8
For z = 4, we get x = 15
For z = 5, we get x = 24
As we can see, for z = 2, the value of x is the smallest. So, the least number of years he must wait for his age to become the cube of a number again is:
(3² + 1) - (2² + 1) = 9 - 4 = 5 years
After 5 years, his age will be 25 which is equal to 5³. Therefore, the correct option is B) 38.
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