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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
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.
Other useful links
- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA
Related Program
I.S.I. & C.M.I. Entrance Program
Taught by students and alumni of I.S.I. & C.M.I.