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13. Solve the congruence: 341x = 2941 (mod 9). v Hint First reduce each number modulo...
Solve the congruence equation arb(mod 13). b-7 d 1. 11. d-9 1, b 8, d9 4, b6 d a2, b6 dm4 5, b8d Soue the con...
Solve the congruence equation arb(mod 13). b-7 d 1. 11. d-9 1, b 8, d9 4, b6 d a2, b6 dm4 5, b8d Soue the congruente eguation axblmod 13) d a a Ti a V) a 4 v) a P vi a- s
Solve the congruence equation arb(mod 13). b-7 d 1. 11. d-9 1, b 8, d9 4, b6 d a2, b6 dm4 5, b8d
Soue the congruente eguation axblmod 13) d a a Ti a V) a 4...
List all points (x,y) in the elliptic curve y2≡ x3 + 2x - 9 (mod 19). (Hint: Corresponding to any...
List all points (x,y) in the elliptic curve y2≡ x3 + 2x - 9 (mod
19). (Hint: Corresponding to any given x , points (x,y) and (x,-y)
can exist on the elliptic curve only if y2≡ x3 + 2x - 9 (mod 19) is
a quadratic residue mod 19. Recall that a value v
∊ Zp is a quadratic residue modulo p only if v(p-1)/2≡ 1 (mod p).
If v is indeed a quadratic residue, we can calculate the two...
Arrange the steps in the correct order to solve the system of congruences x 2 (mod...
Arrange the steps in the correct order to solve the system of congruences x 2 (mod 3), x 1 mod 4). and x3 (mod 5) using the method of back substitution Rank the options below Thus, x= 31.2 - 3/4 + 1)2 - 120+5 We substitute this into the third congruence to obtain 12.5 13 mod 5), which implesu li imod 5) Hence, w5v4 and so x 12.5 - 12/5 + 4) - 5 - 60v. 53, where vis an...
6.32 Theorem. If k and n are natural numbers with (k, d(n)) =I, then there exist positive integers u and v satisfving k...
6.32 Theorem. If k and n are natural numbers with (k, d(n)) =I, then there exist positive integers u and v satisfving ku=(n)u The previous theorem not only asserts that an appropriate exponent is always availahle, but it also tells us how to find it. The numbers u and are solutions lo a lincar Diophantine cquation just like those we studied in Chapter 6.33 Exercisc. Use your observations so far to find solutions to the follow ing congruences. Be sure...
please prove proofs and do 7.4 7.2 Theorem. Let p be a prime, and let b...
please prove proofs and do
7.4
7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
Linear Congruence Generator Below is a generator desigred to simulate rolling a six sided de (d6)....
Linear Congruence Generator Below is a generator desigred to simulate rolling a six sided de (d6). Note: the +1 is to adjust the outcome so you can get a roll of 6 and and do not get o roll of roll, = [(13. rollm-L 4)mod 6 11 5. Using the formula above, determine the outcome of the first 10 rolls in the sequence if the first roll is 3 Roll Number Die Result 6. Come up with a recurrence relation...
Count the number of 6-digit numbers with: Strictly ascending digits from left to right. For example...
Count the number of 6-digit numbers with:
Strictly ascending digits from left to right.
For example 124579. Hint: any 6 distinct digits can be arranged in
ascending order in a unique way.
- The answer is 84
Ascending digits from left to right. For example 133588,
566666, 788888. (hint: choose 6 digits out of {1,2,⋯,9} with
repetition).
What ascending means is that each number is at least the
previous.
- Please solve the second part of the question.
Ascending digits...
Divisibility by 9 To determine if a number is evenly divisible by 9, simply add all...
Divisibility by 9 To determine if a number is evenly divisible by 9, simply add all of the digits in that number. If the sum is divisible by 9, then the original number also is divisible by 9. For example, notice that all of the multiples of 9 from 1 through 12 (9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, and 108) contain digits which sum to 9 or 18 in every case. In another example, to...
Problem 3.1 If each of the 10 digits 0-9 is chosen at random, how many ways...
Problem 3.1 If each of the 10 digits 0-9 is chosen at random, how many ways can you choose the following numbers? (a) A two-digit code number, repeated digits permitted. (b) A three-digit identification card number, for which the first digit cannot be a 0. Repeated digits permitted. (c) A four-digit bicycle lock number, where no digit can be used twice (d) A five-digit zip code number, with the first digit not zero. Repeated digits permitted.
Credit card numbers adhere to certain constraints. First, a valid credit card number must have between...
Credit card numbers adhere to certain constraints. First, a valid credit card number must have between 13 and 16 digits. Second, it must start with one of a fixed number of valid prefixes : 1 • 4 for Visa • 5 for MasterCard • 37 for American Express • 6 for Discover cards In 1954, Hans Peter Luhn of IBM proposed a “checksum” algorithm for validating credit card numbers . 2 The algorithm is useful to determine whether a card...
Solve the congruence equation arb(mod 13). b-7 d 1. 11. d-9 1, b 8, d9 4, b6 d a2, b6 dm4 5, b8d Soue the congruente eguation axblmod 13) d a a Ti a V) a 4 v) a P vi a- s
Solve the congruence equation arb(mod 13). b-7 d 1. 11. d-9 1, b 8, d9 4, b6 d a2, b6 dm4 5, b8d
Soue the congruente eguation axblmod 13) d a a Ti a V) a 4...
List all points (x,y) in the elliptic curve y2≡ x3 + 2x - 9 (mod
19). (Hint: Corresponding to any given x , points (x,y) and (x,-y)
can exist on the elliptic curve only if y2≡ x3 + 2x - 9 (mod 19) is
a quadratic residue mod 19. Recall that a value v
∊ Zp is a quadratic residue modulo p only if v(p-1)/2≡ 1 (mod p).
If v is indeed a quadratic residue, we can calculate the two...
Arrange the steps in the correct order to solve the system of congruences x 2 (mod 3), x 1 mod 4). and x3 (mod 5) using the method of back substitution Rank the options below Thus, x= 31.2 - 3/4 + 1)2 - 120+5 We substitute this into the third congruence to obtain 12.5 13 mod 5), which implesu li imod 5) Hence, w5v4 and so x 12.5 - 12/5 + 4) - 5 - 60v. 53, where vis an...
6.32 Theorem. If k and n are natural numbers with (k, d(n)) =I, then there exist positive integers u and v satisfving ku=(n)u The previous theorem not only asserts that an appropriate exponent is always availahle, but it also tells us how to find it. The numbers u and are solutions lo a lincar Diophantine cquation just like those we studied in Chapter 6.33 Exercisc. Use your observations so far to find solutions to the follow ing congruences. Be sure...
please prove proofs and do
7.4
7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
Linear Congruence Generator Below is a generator desigred to simulate rolling a six sided de (d6). Note: the +1 is to adjust the outcome so you can get a roll of 6 and and do not get o roll of roll, = [(13. rollm-L 4)mod 6 11 5. Using the formula above, determine the outcome of the first 10 rolls in the sequence if the first roll is 3 Roll Number Die Result 6. Come up with a recurrence relation...
Count the number of 6-digit numbers with:
Strictly ascending digits from left to right.
For example 124579. Hint: any 6 distinct digits can be arranged in
ascending order in a unique way.
- The answer is 84
Ascending digits from left to right. For example 133588,
566666, 788888. (hint: choose 6 digits out of {1,2,⋯,9} with
repetition).
What ascending means is that each number is at least the
previous.
- Please solve the second part of the question.
Ascending digits...
Problem 3.1 If each of the 10 digits 0-9 is chosen at random, how many ways can you choose the following numbers? (a) A two-digit code number, repeated digits permitted. (b) A three-digit identification card number, for which the first digit cannot be a 0. Repeated digits permitted. (c) A four-digit bicycle lock number, where no digit can be used twice (d) A five-digit zip code number, with the first digit not zero. Repeated digits permitted.
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