To solve the problem, we need to identify the age of Mr. X last year that satisfies two conditions: it must be the square of a number last year and the cube of a number next year. Then, we will determine the minimum number of years Mr. X must wait for his age to become the cube of a number again.
### Step-by-Step Solution:
1. Identify Mr. X's Age Last Year and This Year:
- We need to find the smallest age [tex]\( n \)[/tex] such that:
- [tex]\( n^2 \)[/tex] is Mr. X's age last year.
- [tex]\( n^2 + 2 \)[/tex] is the cube of a number.
2. Mathematical Representation:
- Let [tex]\( n \)[/tex] be an integer. Then:
- Mr. X's age last year = [tex]\( n^2 \)[/tex]
- Mr. X's age this year = [tex]\( n^2 + 1 \)[/tex]
- Mr. X's age next year = [tex]\( n^2 + 2 \)[/tex]
3. Condition for Next Year’s Age:
- Mr. X’s age next year must be a perfect cube.
- We need to find the smallest [tex]\( n \)[/tex] such that [tex]\( (n^2 + 2) \)[/tex] is a perfect cube.
4. Finding the Valid Age:
- By examining small values of [tex]\( n \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
- [tex]\( n^2 + 2 = 1 + 2 = 3 \)[/tex] (not a cube)
- For [tex]\( n = 2 \)[/tex]:
- [tex]\( n^2 + 2 = 4 + 2 = 6 \)[/tex] (not a cube)
- Continue this process until:
- For [tex]\( n = 5 \)[/tex]:
- [tex]\( n^2 + 2 = 25 + 2 = 27 \)[/tex]
- [tex]\( 27 \)[/tex] is indeed [tex]\( 3^3 \)[/tex], a perfect cube.
- Therefore, Mr. X's age last year was [tex]\( 5^2 = 25 \)[/tex] years old.
- Mr. X's age next year is [tex]\( 27 \)[/tex], a perfect cube.
5. Finding Number of Years until Mr. X's Age is a Cube Again:
- We start from his current age (which is 26 years) and increment year by year, checking when it will next be a cube.
- From the solution, we found the next instance where Mr. X's age is a cube occurs 37 years after his current age.
So, the least number of years Mr. X must wait to become the cube of a number again is:
Answer: (B) 38
It results in the next occurrence of a perfect cube age after 27 being precisely 27 years added to 10 giving - Corrected final answer is: 27 + 37. Thus answer finalized:B. 38 years.
