1. Compute the following values. Here the operation mod n is the remainder opera
ID: 2902748 • Letter: 1
Question
1. Compute the following values. Here the operation mod n is the remainder operation as defined in the book. Recall that 0 <= r < n where r represents the remainder and m represents the modulus.
a) 15 mod 7=______
b) 10 mod 7=______
c) 150 mod 7 = _____
d) 2^6 mod 5=_____
e) 2^4 mod 5=_____
f) 2^10 mod 5=_____
g) 100 mod 3=____
h)13 mod 3=_____
i) 1300 mod 3=____
j) If p mod 7= 4 and q mod 7= 3, then p*q mod 7 is in what equivalence class of congruence mod 7?
2. Compute the following values. Here the operation mod n is the remainder operation as defined in the book. Recall that 0 <= r < n where r represents the remainder and m represents the modulus.
a) 50 mod 7= ______ and 15 mod 7 = ________
b) 10 mod 7= ______ and 3 mod 7= _________ note that 50=10*5 and 15=3*5
c) 24 mod 11= _____ and 68 mod 11=________
d) 6 mod 11= _____ and 17 mod 11=________ note that 24=6*4 and 68=17*4
e) 12 mod 5=______ and 27 mod 5= ________
f) 4 mod 5=______ and 9 mod 5 = ________ note that 12=4*3 and 27=9*3
g) 30 mod 3=______ and 45 mod 3=________
h) 6 mod 3=______ and 9 mod 3= _________ note that 30=6*5 and 45=9*5
i) 52 mod 7=_______ and 80 mod 7=________
j) 13 mod 7=_______ and 20 mod 7=________ note that 52=13*4 and 80=20*4
3. The computations in question 2 are organized in 5 pairs a-b, c-d, e-f, g-h and i-j. Using these pairs as observation, which of the following statements appear to be true. Assume n represents the modulus, k represents a common factor and a,b are integers.
a) True or False If n and k are relatively prime and if ak mod n=bk mod n, then a=b.
b) True or False If n and k are relatively prime and if ak mod n=bk mod n, then ak=bk.
c) True or False If n and k are relatively prime and if ak mod n=bk mod n, then a mod n=b mod n.
d) True or False If n and k are relatively prime and if ak mod n=bk mod n, then a mod n=ak mod n.
4. See page 128 problem 22 for information relating to this question. Guess and check is an acceptable technique for these problems.
5. Find an integer value for x if possible, that makes the following statements true.
6. Using the computations in problem 5 as examples, find an integer value for x if possible, that makes the following statements true. (k represents a nonzero integer.)
1. Compute the following values. Here the operation mod n is the remainder operation as defined in the book. Recall that 0Explanation / Answer
1. a) 15 mod 7= 1
b) 10 mod 7=3
c) 150 mod 7 = 3
d) 2^6 mod 5=4
e) 2^4 mod 5=1
f) 2^10 mod 5=4
g) 100 mod 3=1
h)13 mod 3=1
i) 1300 mod 3=1
j) If p mod 7= 4 then p=7*N +4 and q mod 7= 3, then q=7*N +3
j) If p mod 7= 4 then p=7*N +4 and q mod 7= 3, then q=7*N +3
p*q mod 7 =5
2. a 1
b. 3
c. 2
d. 6
e. 2
f. 4
g. 0
h. 0
i. 3
j. 6
3. a. false
b. false
c. true
d. false
4. a. 2x= 3mod8
2x = 8k+3
x = 4k+1.5 so x will not be integer
b. 5x = 6 mod8
5x = 8k+6
x = 1.6k+1.2 at k=3 so x=6
5. a. 6
b. 0
c. 3
d. 6
e. 0
f. 3
g. 3
h. 0
6. a. x=0 if k = 1
b. x=6 if k=6
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