Answer :
(a) [tex]\varphi(13)=12[/tex] since 13 is prime.
(b) [tex]81=3^4[/tex], and there are 81/3 = 27 multiples of 3 between 1 and 81, which leaves 81 - 27 = 54 numbers between 1 and 81 that are coprime to 81, so [tex]\varphi(81)=54[/tex].
(c) [tex]100=2^2\cdot5^2[/tex]; there are 50 multiples of 2, and 20 multiples of 5, between 1 and 100; 10 of these are counted twice (the multiples of 2*5=10), so a total of 50 + 20 - 10 = 60 distinct numbers not coprime to 100, leaving us with [tex]\varphi(100)=100-60=40[/tex].
(d) [tex]102=2\cdot3\cdot17[/tex]; there are 51 multiples of 2, 34 multiples of 3, and 6 multiples of 17, between 1 and 102. Among these, we double-count 17 multiples of 2*3=6, 3 multiples of 2*17=34, and 2 multiples of 3*17=51; we also triple-count 1 number, 2*3*17=102. There are then 51 + 34 + 6 - (17 + 3 + 2) + 1 = 70 numbers between 1 and 102 that are not coprime to 102, and so [tex]\varphi(102)=102-70=32[/tex].