The positive rational numbers form a group under a * b = ab/10. Which group axiom fails if the set is narrowed to the positive integers?

The positive rational numbers form a group under $a * b = \frac{ab}{10}$ . Which group axiom fails if the set is narrowed to the positive integers?

Aclosure, since ab/10 need not be whole

Bidentity, since 10 is not a positive integer

Cassociativity, since division is not associative

Dcommutativity, since ab is not always ba

Explanation

Take

a=b=3

: both are positive integers but

3*3=\dfrac{9}{10}

is not, so the set is not closed. The other axioms survive, since

(a*b)*c=\dfrac{abc}{100}=a*(b*c)

ab=ba

makes the operation commutative, and 10 is a positive integer.

Derived from ZIMSEC Further Mathematics 9187/1, November 2010, Q4

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