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Dear Hemanthkrishna,applying the concept of mod we get
5^2 is congruent to 3 mod 11 [ means when 3 is subtracted from 5^2=25 it is divisible from 11 ]raising to 5th power... 5^10 is congruent to 3^5 mod 11but 3^5=243 is congruent to 1 mod 11hence 5^10 is congruent to 1 mod 11raising again to 4th power... 5^40 is congruent to 1 mod 11HENCE 5^40 = 11a + 1So N=1Similarly for 2^2003WE have.. 2^4 is congruent to -1 mod 17raising to 500 power.. 2^2000 congruent to 1 mod 17Hence 2^2000=17b + 1Multiplying by 2^3 we have2^2003 = 17.b.8 + 82^2003 = 17c + 8Hence M=8SO M-N=8-1=7(ANS)Regards.
5^2 is congruent to 3 mod 11 [ means when 3 is subtracted from 5^2=25 it is divisible from 11 ]raising to 5th power... 5^10 is congruent to 3^5 mod 11but 3^5=243 is congruent to 1 mod 11hence 5^10 is congruent to 1 mod 11raising again to 4th power... 5^40 is congruent to 1 mod 11HENCE 5^40 = 11a + 1So N=1Similarly for 2^2003WE have.. 2^4 is congruent to -1 mod 17raising to 500 power.. 2^2000 congruent to 1 mod 17Hence 2^2000=17b + 1Multiplying by 2^3 we have2^2003 = 17.b.8 + 82^2003 = 17c + 8Hence M=8SO M-N=8-1=7(ANS)
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