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Integer and Divisibility | B.Stat Objective | TOMATO 69

Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Integer and Divisibility. You may use sequential hints.

Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Integer and Divisibility.

Integer and Divisibility (B.Stat Objective problems)


Every integer of form \((n^{3}-n)(n-2)\) for n=3,4,….. is

  • divisible by 12 but not always divisible by 24
  • divisible by 6 but not always divisible by 12
  • divisible by 24 but not always divisible by 48
  • divisible by 9

Key Concepts


Logic

Integers

Divisibility

Check the Answer


But try the problem first…

Answer: divisible by 6 but not always divisible by 12

Source
Suggested Reading

B.Stat Objective Question 69

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

\((n^{3}-n)(n-2)=n(n^{2}-1)(n-2)=(n-1)n(n+1)(n-2)\)

Second Hint

(n-1)n(n+1) is divisible by 3 and any two consecutive integers is divisible by 2 gcd(2,3)=1

Final Step

then 6|(n-1)n(n+1) and minimum (n-2)=1 for n=3,4,…. then \((n^{3}-n)(n-2)\) divisible by 6 but not always divisible by 12.

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