Write true (T) or false (F) for the following statements: (i)392 is a perfect cube.
(ii)8640 is not a perfect cube.
(iii)No cube can end with exactly two zeros.
(iv)There is no perfect cube which ends in 4.
(v)For an integer a, `a^3` is always greater than `a^2dot`
(vi)If `a\ a n d\ b` are integers such that `a^2> b^2, t h e n\ a^3> b^3dot`
(vii)If `a` divides `b` , then `a^3` divides `b^3dot`
(viii)If `a^2` ends in 9, then `a^3` ends in 7
(ix)If `a^2` ends in 5, then `a^3` ends in 25.
(x)If `a^2` ends in an even number of zeros, then ends in and odd number zeros.

Explanation

(i) `392` is a perfect cube.
By prime factorization,
`392=2xx2xx2xx7xx7=23xx72`
Hence the statement is False.

(ii) `8640` is not a perfect cube.
By prime factorization,
`8640=2xx2xx2xx2xx2xx2xx3xx3xx3xx5=23xx23xx33xx5`
Hence the statement is True.

(iii)No cube can end with exactly two zeros.
Reason: Because a perfect cube always have zeros in multiple of `3`.
Hence the statement is True.

(iv) There is no perfect cube which ends in `4`.
Reason: `4xx4xx4` and it ends with `4`.
Hence the statement is False.

(v) For an integer `a`, `a^3` is always greater than `a^2`.
Reason: Because in case of negative integers,
`(-2)^2=4` and `(-2)^3=-8`
Hence the statement is False.

(vi) If `a` and `b` are integers such that `a^2>b^2`, then `a^3>b^3`.
Reason: In case of negative integers, `(-5)^2>(-4)^2=25>16`.
But, `(-5)^3>(-4)^3=-125>-64` is not true.
Hence the statement is False.

(vii) If `a` divides `b`, then `a^3` divides `b^3`.
Reason: If `a` divides `b=b/a=k`, so `b = ak`
`= b^2/a^2=(ak)^3/a^3=a^3k^3/a^3=k^3`
For each value of `b` and `a` its true.
Hence the statement is True.

(viii) If `a^2` ends in `9`, then `a^3` ends in `7`.
Reason: Let `a=7`
`7^2=49` and `7^3=343`
Hence the statement is False.

(ix) If `a^2` ends in an even number of zeros, then `a^3` ends in `25`.
Let `a=20`
`=a^2=20^2=400` and `a^3=8000`
Hence the statement is False.

(x) If `a^2` ends in an even number of zeros, then `a^3` ends in an odd number of zeros.
Reason: Let `a=100`
`=a^2=100^2=10000` and `a^3=100^3=1000000`
Hence the statement is False.