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# Find the remainder when :121^n-25^n+1900^n-(-4^n) /2000

SHAIK AASIF AHAMED
6 years ago
Hello student,
We know that 2000 = 16*25,
hence we prove that both 16 and 25 divide it making the use of fact that (a-b) divides (an – bn).
16 divides 121n-25n +1900n-(-4n):
Now, (121-25) / (121n-25n) ⇒ 96/ (121n-25n) ⇒ 16/ (121n-25n) Now, (1900-(-4)) / 1900n-(-4n) ⇒ 1904/ 1900n-(-4n) ⇒ 16/ (1900n-(-4n))
Similarly we proceed to show that 125 divides 121n-25n +1900n-(-4n):
Now, (121-(-4)) / (121n-(-4n)) ⇒ 125/ (121n-(-4n))
Now, (1900-25) / 1900n-25n) ⇒ 1875/ (1900n-25n) ⇒ 125/ (1900n-25n)
Hence, we have proved that both 16 and 125 divide 121n-25n +1900n-(-4n) which means that 2000 divides 121n-25n +1900n-(-4n).