2023 · Q81
Suppose a, b and c are three distinct natural numbers such that a + b + c = abc.
Consider the following statements:
- 1The arithmetic mean of a, b and c is a natural number.
- 2The harmonic mean of a, b and c lies between 1 and 2.
Which of the statements given above is/are correct?
general-mental-abilityquantitative-aptitude
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Let the numbers be ordered a < b < c. With a = 1, the equation 1+b+c = bc gives (b-1)(c-1) = 2, yielding the unique solution {1, 2, 3} in distinct natural numbers. Their arithmetic mean is (1+2+3)/3 = 2, a natural number, so statement 1 is correct. Their harmonic mean is 3/(1 + 1/2 + 1/3) = 18/11, about 1.64, which lies between 1 and 2, so statement 2 is correct. Hence both statements are correct.