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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

Logic

Integers

Divisibility

## Check the Answer

But try the problem first…

Answer: divisible by 6 but not always divisible by 12

Source

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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