CMI2016-A-2

Question

The symbol $\mid$ reads as "divides", and $\nmid$ as "does not divide". For instance, $2 \: \mid \:6$ and $2 \: \nmid \: 5$ are both true. Consider the following statements.

1. There exists a positive integer $a$ such that $(2 \mid (a^3 -1))$ and $( 2 \mid a)$.
2. There exists a positive integer $b$ such that $6 \nmid (b^3 -b)$.

What can you say about these statements?

• A. Only i is true
• B. Only ii is true
• C. Both i and ii are true
• D. Neither i nor ii is true

Comments

@Arjun sir, the question is mistyped.

Statement 1 is "There exists a positive integer a such that (2 | ($a^{3}$ − 1)) and (2 | a)."

https://www.cmi.ac.in/admissions/sample-qp/pgcs2016.pdf

Answer 1

i. is False because 2 does not divides 1 ("and" is used so no need to check for 2 divides $(a^3-1)$)

ii. $(b^3-b) = b(b^2-1) = b(b-1)(b+1) = (b-1)b(b+1)$
multiplication of three consecutive numbers
means one of them must be even making whole multiplication even and also one of them must be divisible by 3
an even number divisible by 3 is also divisible by 6
which makes ii statement False

so ans is option D

Comments

Option B true

Answer 2