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CAT 2001 — QA Question 41

Number TheoryEasy
Passage / Data

Answer the following question based on the information given below.

The batting average (BA) of a test batsman is computed from runs scored and innings played-completed innings and incomplete innings (not out) in the following manner:

r1 = number of runs scored in completed innings; n1 = number of completed innings

r2 = number of runs scored in incomplete innings; n2 = number of incomplete innings

BA = r1+r2n1

To better assess batsman's accomplishments, the ICC is considering two other measures MBA1 and MBA2 defined as follows:

MBA1 = r1n1+n2n1× max[0, (r2n2-r1n1])

MBA2 = r1+r2n1+n2

Let b be a positive integer and a = b2 – b. If b ≥ 4, then a2 – 2a is divisible by

Answer & solution

  • A

    15

  • B

    20

  • 24

  • D

    None of these

Solution

a2 – 2a = a(a – 2)

Substituting a = b2 – b, we get,

a2 – 2a = (b2 – b)[(b2 – b) – 2]

             = b(b – 1)(b2 – b – 2)

             = b(b – 1)(b – 2)( b + 1)

             = (b – 2)(b – 1)b(b + 1)

This is a product of 4 consecutive positive integers which will definitely have a multiple of 2, 3 and 4.

∴ This must be divisible by 2 × 3 × 4 = 24

Hence, option (c).

CAT 2001 QA Q41: Let b be a positive integer and a = b 2 – b. If b ≥ 4, then a 2 – 2a is divisible by — Solution | TheCATExam