Determine whether the following operations define a binary operation on the given set or not: ' _6' on S = 1, 2, 3, 4, 5 defined by, a _6 b = Remainder when ab is divided by 6.

Consider the composition table, _6 1 2 3 4 5 1 1 2 3 4 5 2 2 4 0 2 4 3 3 0 3 0 3 4 4 2 0 4 2 5 5 4 3 2 1 Here all the elements of the table are not in S. For a = 2 and b = 3, a _6b= 2 _63 = remainder when 6 divided by 6=0 Thus, _6 is not a binary operation on S.

Determine whether the following operations define a binary operat…

Question

Determine whether the following operations define a binary operation on the given set or not:
$'\times _6'$ on $S = \{1, 2, 3, 4, 5\}$ defined by, $a \times _6 b =$ Remainder when ab is divided by $6.$

$\times _6$	$1$	$2$	$3$	$4$	$5$
$1$	$1$	$2$	$3$	$4$	$5$
$2$	$2$	$4$	$0$	$2$	$4$
$3$	$3$	$0$	$3$	$0$	$3$
$4$	$4$	$2$	$0$	$4$	$2$
$5$	$5$	$4$	$3$	$2$	$1$

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