From the four options given below, choose the one which best describes the…
2019
From the four options given below, choose the one which best describes the relationship among the following three numbers: 343 : 1331 : 2197
- A.
These numbers are squares of odd numbers.
- B.
These numbers are square roots of numbers.
- C.
These numbers are cubes of odd numbers.
- D.
These numbers are natural numbers.
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Show answer & explanation
Correct answer: C
Concept: A perfect cube is a number of the form n3, the product of an integer with itself three times. To classify a set of numbers by a hidden pattern, test them against the basic powers — squares (n2), cubes (n3) and so on — and pick the most specific property that ALL of them share.
Application: Express each of the three numbers as a power of a small integer:
343 = 73
1331 = 113
2197 = 133
Each value is exactly the cube of an integer, and the bases 7, 11 and 13 are all odd. So the property shared by all three is: cubes of odd integers.
Cross-check: Compare against the rejected descriptions, by value:
Squares: 343 lies between 182 = 324 and 192 = 361, so it is not a perfect square; the same is true for 1331 and 2197. The squares description fails.
Square roots of numbers: every number is the square root of its own square, so this label fits any number and identifies no pattern at all.
Natural numbers: true but trivial — it is the weakest possible description and ignores the cube structure.
Note: 7, 11 and 13 are also prime, so these are cubes of primes too; among the four choices offered, however, only cubes of odd numbers is listed, and it correctly fits all three values.