Practice test 2 – CS 1050C – solutions
1. Notice that (all congruences below are modulo 4):
(cncn−1. . . c1c0)10 =c0+ 10c1+ 102c2+ 103cc+. . . + 10ncn≡c0+ 2c1.
The congruence holds because 10 ≡2 and 10n≡0 for n≥2. We
conclude that (cncn−1. . . c1c0)10 is divisible by 4 (i.e. congruent to
zero modulo 4) if and only if c0+ 2c1is. 2
2. Notice that
gcd(n, n + 12) = gcd(n, n + 12 −n) = gcd(n, 12).
This proves that gcd(n, n + 12) is a divisor of 12 for any n, i.e. it is
an element of the set A={1,2,3,4,6,12}. Conversely, any element
a∈Ais a value of gcd(n, n + 12) for some n, for example for n=a
(because gcd(a,12) = a as ais a divisor of 12).
The list of all possible values is 1,2,3,4,6,12.
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3. We’ll do it directly from definitions here.
First, let’s prove that gis onto. Take any c∈C. Then, h(c) is an
element of Dso it has to belong to the image of (onto function by
assumptions) h◦g◦f: there exists an a∈Asuch that h◦g◦f(a) =
h(g(f(a))) = h(c). Because his a one-to-one function, g(f(a)) = c
(value of hon both sides is the same!). We conclude that cis in the
range of g(it’s the value of gat f(a)). We are done.
Now we need to prove that gis one-to-one. Assume it is not. Then,
there are b, b0∈Bsuch that b6=b0and g(b) = g(b0). Since fis onto,
there are a, a0∈Asuch that f(a) = band f(a0) = b0. Obviously, a6=a0
(otherwise, the values of fat these elements would be the same but
they happen to be band b0). Moreover,
h◦g◦f(a) = h(g(f(a)) = h(g(b)) = h(g(b0)) = h(g(f(a0))) = h◦g◦f(a0).
This contradicts one-to-oneness of h◦g◦f.2
4. Assume a, b, c are natural numbers and x, y are integers such that xa +
yb =c. gcd(a, b) is (by definition) a common divisor of aand b:
gcd(a, b)|aand gcd(a, b)|b. Thus, gcd(a, b) divides any integer
combination of aand b, in particular xa +yb =c2
5. Notice that, since ais even,
gcd(a, a + 2) = gcd(a, 2) = 2
(this is because ais an even number so 2 is a common divisor. Clearly,
there are no larger divisors). Now, from one of the theorems we proved
in class,
2lcm(a, a + 2) = gcd(a, a + 2)lcm(a, a + 2) = a(a+ 2).
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