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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.

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


Answer: divisible by 6 but not always divisible by 12

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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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


Answer: divisible by 6 but not always divisible by 12

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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