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Prove that (i) 3^37 + 37 = 0 (mod 100) (ii) 3^33 + 77 = 0 (mod 100)

Prove that


 


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


 


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

Grade:10

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
9 years ago
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 337/4=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)

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