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Discrete Mathematics with Graph Theory (3rd Edition) 178

# Discrete Mathematics with Graph Theory (3rd Edition) 178 -...

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176 Solutions to Exercises 2. Let the integers be aI, a2, ... , a6 and consider the six integers all al + a2, ... , al + ... + a6. If one of these is divisible by 6, the result holds. Otherwise, there are only five possible remainders when each of these is divided by 6 (namely, 1, 2,3,4 or 5). The Pigeon-Hole Principle says that two of these must give the same remainder; so for some s =f. t, al + a2 + ... + as = 6q + r al + a2 + ... + at = 6q' + r for the same remainder r. Assuming s > t (without loss of any generality), subtracting gives at+l + ... + as = 6( q - q') and the result holds. 3. Let the integers be aI, a2, ... ,an and consider the n integers aI, al + a2, ... , al + ... + an. If one of these is divisible by n, the result holds. Otherwise, there are only n - 1 possible remainders when each of these is divided by n (namely, 1,2, ... , or n - 1). The Pigeon-Hole Principle says that two of these must give the same remainder; so for some s =f. t, al + a2 + ... + as al + a2 + ... + at nq+r nq' + r for the same remainder r. Assuming s > t (without loss of any generality), subtracting gives at+l + ... + as = n(q - q') and the result holds.
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