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Midterm Exam Solution

# Prove by induction qn 5n 1 50 1 2 q0 which

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Unformatted text preview: induction: Q(n) = 5n + 1 50 + 1 = 2 = Q(0), which demonstrates that this is true for n = 0. 51 + 1 = 6 = Q(1) (also true for n = 1). We assume that Q(n-1) holds And that Q(n-2) holds Q(n) = 6(5n-1 + 1) – 5(5n-2+ 1) = 6*5n-1 - 5n-1 + 6 –5 = 5*5n-1 + 1 = 5n + 1. 5n-1 + 1 5n-2 + 1 3b) Let n = n6n5n4n3n2n1n0 for some decimal number n. Prove that 11|n if and only if 11| (n0 – n1 + n2 - n3 + n4 – n5 + n6). n = d0 + 10 * d1 + 100 * d2 + … + 10kdk. Alternate sum of the digits of number n is: dk – dk-1 + dk-2 - … + (-1)kd0 Let us consider the following expression: E = (10 + 1) * (d0 + 10(d1 – d0) + 100(d2 – d1 – d0) + … + 10k(dk – dk-1 + dk-2 - … + (-1)kd0)) – 10k+1(dk – dk-1 + dk-2 - … + (-1)kd0). After performing careful multiplication, one can see that E = n. A term that has a factor of (10 + 1) or 11, is divisible by 11. Thus n is divisible by 11 whenever 10k+1(dk – dk-1 + dk-2 - … + (-1)kd0) is divisible by 11. Since 10k+1 is not divisible by 11 and 11 is a prime, we can make a conclusion that n is divisible by 11 if and only if dk – dk-1 + dk-2 - … + (-1)kd0 is divisible by 11. This number is the alternate sum of digits of n....
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