If a # 0 (mod m) and b # 0 (mod m), then ab# 0 (mod m). Select one: O True False

Q: If m is an integer, and if (m mod 11) = 6, then what is (5m mod 11)? 2 4 11

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Q: . If 9x = 15 (mod 37), then x mod 37 is 14 17 33 11

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Q: If a mod b = 0, then b | a O True O False

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Q: If a = b (mod m) and a = b (mod n) then a = b (mod mn)

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Q: If a= b (mod n) and gcd (a, n) = d, then gcd (b, n) = d :Select one True False

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Q: If cy (mod n) andc>0 then a) x= y (mod cn) b) x y (mod n) ) xy d) (mod") None of these

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Q: 2. If x mod 5 = 4, then x could be O 571 O 400 O 633 O 719

A: Given               x(mod 5)=4

Q: 9. Let a and b be integers. Prove that if a =7 (mod 8) and b = 3 (mod 8), then: (a) a + b = 2 (mod…

A: Your questions solution below in images

Q: If a≡b (mod m) and c ≡d (mod m),show that (b) ac≡bd (mod m).

A: Given : a≡b (mod m) c ≡d (mod m), We have to show  ac≡bd (mod m)

Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b = c (mod n) (b) b2 =…

A: We use the properties of congruences to answer the given question.

Q: Find 12 ^– 1(mod 17), the inverse of 12 modulo 17

A: Find an inverse of 12 modulo 17 as follows.

Q: mod

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Q: If a =b (mod m) and a = b (mod n) then a = b (mod m+n) Select one: O True O False

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Q: 3. Show that ifn = 2 (mod 4), then 9" + 8" is divisible by 5.

A: Here we will prove the given statement.

Q: If 9x = 15 (mod 37), then x mod 37 is %3D 33 11 14 O 17

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Q: Prove or disprove If ac = bc(mod m), then a = b(mod m). If n | m and a = b(mod m), then a = b(mod…

A: According to the given information, it is required to prove or disprove the given statement:

Q: Given a = 4( mod 9) and b = 5( mod 9), find c with 0<c<9 such that c = 5a + 3b ( mod 9)

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Q: The remainder in 14 (mod 3) is 3

A: Given that 14 (mod3)

Q: a) If a| (21b−c) and 6c≅3b(mod2a), then a| 41c b) If a≅b(modc) and d|c then ?≅?(mod?) c) ∴(d|…

A: this is a question of modular arithmatic.

Q: If a = b (mod n) and b =c (mod n) then which of the following is false. a = c (mod n) a = c3(mod n)…

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Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

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Q: you Prove the " of 25nY-18n +7 =0 (mod4), then statement contradiction. is by "even

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Q: 3 23) Give a counter example to disprove the foilowing Property ab = ac (modn) →6=C (modn) yb=C…

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Q: If a = 1 (mod n) and b = 1 (mod n) then a - b = 0 (mod n) اختر واحدا: 2 3

A: True.

Q: If (a, 42) =1, then a^6-2=?mod 168 1 -1 None of these

A: Just apply the Fermat's Theorem.

Q: In the formula a b (mod n), what is the value of b in 36 b (mod 5)?

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Q: Which of the following are true modulo 12? (1) Does 5z = 6 mod 12 have a solution? How many? (2)…

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Q: (18) If a, be Z, then (a + b)3 = a3 + b3 (mod 3).

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Q: If n is an integer, then (2n² + 3n)# 3(mod5).

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Q: State as TRUE or FALSE. d) 143 = -40 (mod 5) e) –4 = 788 (mod 3)

A: a≡b(mod n) means a-b is divisible by n

Q: Suppose that a = 3 mod 12 and b =7 mod 12 IfC = 3 a – 2 b mod 12 then C =

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Q: 2) If a? = b² (mod n), then a = b (mod n) a) True b) False O a) True O b) False

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Q: The order of 2 mod 17 is: a) 2 b) 4 c) 8

A: The order of a modulo p then is the least positive integer n such that an ≡ 1(mod p) where p is a…

Q: (c) For all integers a and b, if a · b = 1 (mod 6), then a =1 (mod 6) or b = 1 (mod 6). (d) For all…

A: (c) The statement is false. For example take a=5, b=5 ab=2525≡1 mod(6) Hence ab≡1 mod(6) But a≡5…

Q: Ha = b (mod n) and gcd (a,n)= d, then gcd (b,n) %3D

A: Given a≡b(mod n). This can be written as a-b≡0(mod n). Implies a-b=kn for some natural number k.…

Q: aibic, and m are integers such that IF 9,b,c, and m m70, dc, d|m, and ac = bc (mod m), then a =b…

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Q: Prove directly if the statements that are true, give counterexamples to disprove those that are…

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Q: If a+c=b+c (mod m) then a =b (mod m) Select one: True False

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Q: If 5x = 12 (mod 41), then the value of x (mod 41) is O 33 O 19 D 27 O 17

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Q: 3. If x mod 4 = 1 and y mod 4 = 3, then xy could be O 901 O 299 O 508 O 113

A: We shall use the division algorithm

Q: x is congruent to 21 mod 8. Then x is None of the option 29 16 24

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Q: 20. If a e Z and a = 1 (mod 5), then a? = 1 (mod 5).

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Q: Show that if x mod 8 = 1, then (3x −5) mod 6 = 4.

A: Solution

Q: If ab = 0 (mod m), then either a = 0 (mod m) or D=0 (mod m). Select one: O True False

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Q: Does

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Q: 12.ifa b(mod n) and b a c(mod n) then ab m be(mod n)

A: We can proove in the following way:-

Q: If 3(26!) = x(mod29), then x = None of the choices O 11 13 15

A: The given problem is to solve the given congruence equation. 3(26!)= x (mod 29).

Q: 20. If a e Z and a = 1 (mod 5), then a² = 1 (mod 5). If a - h (mod n) then a3 = h3 (mod n)

A: Notation: a|b  means a divides b or                   a|b means b is divisible by a.

Q: 3. Let d = (a, n) where n > 1. Prove that if there is a solution to ax = b (mod n), then d divides…

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Q: (2) If a = b modn and a = b mod m and ged(m, n): 1. then a =b mod mn.

A: The greatest common divisor of two or more integers, which are not all zero, is the largest positive…

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.