Prove that

(i) 3^37 + 37 = 0 (mod 100)

(ii) 3^33 + 77 = 0 (mod 100)

Hello student,
Please find the answer to your question below
1)3³⁷has units digit of 3
because powers of 3 have units digits of 3,9,7,1 in cycles of 4
so 3^37/4=3¹=3 is units digit
To find 10’s digit 3³⁷=3(3)³⁶=3(81)⁹=3*8*9=216 so 6 is tens digit
so 3³⁷+37 has last 2 digits=63+37=100
so3^37 + 37 = 0 (mod 100)
2)similarly 3³³has units digit of 3 and tens digit=3(81)⁸=3*8*8=192
so last 2 digits are 23
last 2 digits of 3^33 + 77=23+77=100
So3^33 + 77 = 0 (mod 100)