Any integer of the form 12k + 11 is also of the form а) Зт + 1 b) 5m + 2 4m + 3 бт + 1

Q: The number 6 has four divisors -- namely: 1, 2, 3, and 6. List all of the numbers less than 20 that…

A: Given that: Divisors of number 6 = 1, 2, 3, 6

Q: Prove that Va - Vb is irrational number if 'a' and 'b’ are prime number?

A: Given To prove that a-b is irrational number if a and b are prime number.

Q: Find all primes that are the difference of the fourth powers of two integers.

A: Let the two integers are x and y and n is prime so we have  n=(x4-y4)       (suppose x>y)…

Q: 2. Prove that 11" – 7" is divisible by 4 for any positive integer п.

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Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108.

A: Note: As per our company guidelines we are supposed to answer the first question only. Kindly ask…

Q: Prove the statement. "For any integer n, nº +4 is not divisible by 8." by Contradiction.

A: Given: n be any integer. To prove: n3+4 is not divisible by 8.

Q: he sum of any two integers of the from 4k + 1isalways: COdd prime Zero even

A: The sum of two consecutive integers can always be written as 2n+1, but it can be written as 4n+1…

Q: Justify that every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q.

A: Let p be a prime number other than 2 and 3. If p is divided by 6, then by the quotient-remainder…

Q: If a, b are two distinct prime number than a highest common factor of a, b is: (2 ab

A: HCF of two numbers is found out by taking common prime factors from both the numbers the…

Q: A natural number greater than 1 that has only itself and 1 as factors is called a/an number.

A: Given that  A natural number greater than 1 that has only itself and 1 as factors is called…

Q: The diagram below shows ten circles joined by five lines with three circles Place all the…

A: Requirements: Numbers should range from 1 to 10.   Every line has three circles.   3. The amount for…

Q: Prove that a number in the form 6k+5,” what I mean is a number in the form of 6 times some integer…

A:

Q: Consider all positive integers that are mueiples of 20 and that are less than or equal to 300. What…

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Q: b. Prove that the integer 359,240,758 cannot be a perfect quartic. (i.e. fourth power) that 250 758…

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Q: Show that every positive integer can be written as the product of a square (possibly 1) and a…

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Q: (a) Show that if seven integers are selected from the first 10 positive integers, there must be at…

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Q: Which positive integers less than 20 are relatively prime to 20?

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Q: Prove that 2^10 + 5^12 is a composite number.

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Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

Q: Find, If possible, A + B, A- B, 2A, 2A - B, and B +A. (If not possible, enter IMPOSSIBLE In any…

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Q: The sum of all positive integers less than 300 which are divisible by 3.

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Q: Prove by cases that n2−1 is divisible by 3 when n is an integer not divisible by 3.

A: The detailed solution is given below Thank you

Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

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Q: if a is a positive integer less than 10, then 7a7,7a7,7a7 is divisible by 3 Select one: O True O…

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Q: "Every positive integer can be written as the sum of the square of 3 positive integers." prove…

A: Given: "Every positive integer can be written as the sum of the square of 3 positive integers." To…

Q: The sum of all positive integers less than 300 which are not divisible by 3.

A: Sum of n positive integer is given by, Sn=nn+12

Q: Suppose a and b are integers that divide the integer c. If a and b arerelatively prime, show that ab…

A: Since a/c   &  b/c  , we know there exists integers K1 & K2  ,with K1 a=K2 b=c--1Also…

Q: Suppose that a, b are two integers such that 3a? – b3 = 1. Does it follow that a and b are…

A: Given that a, b are two integers such that 3a2-b3=1.

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Suppose there is a currency which only uses 3-cent coins and 7-cent coins. Prove that for all…

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Q: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

Q: show that every integer greater than 11 is the sum of two composite integers

A: We need to prove that  every integer greater than 11 is the sum of two composite integers.

Q: Construct four relatively prime integers a, b, c, d such that no three of them are relatively prime.

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Q: 2. Show that if a]b then a'|b² for all integers a and b (Symbols V, 3, A, V, V,A,→,-)

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Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Show that there are no prime triples other than 3, 5, 7.

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Q: The positive integers that are divisors of 68 are

A: We will use multiple method to find all the divisor of the given number .

Q: If a and b are integers with gcd(a, b) =d Then gea (-) -- gcd

A: False

Q: Prove that 2^(27) - 1 is not a prime number.

A: A prime number is a natural number greater than 1 that is not a product of two smaller natural…

Q: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

Q: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

Q: 1. Show that if seven integers are selected from the first 10 positive integers, there must be at…

A: We can solve the following question by pigeonhole principle

Q: How large a subset of the integers from 1 to 50 must be chosen to guarantee that one of the numbers…

A: We have to determine How large a subset of integers from 1 to 50 must be chosen to guarantee that…

Q: Exercise 5. Prove that given positive integers a and b, there is a positive integer d such that • d…

A: Given two positive integers a and b, let us choose d = gcd(a, b) [greatest common divisor] Then…

Q: 5!, called 5_______ , is the product of all positive integers from_________ down through________ .…

A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: 4. Suppose that a and b are integers that divide the integer c. If a and b are relatively prime,…

A: Suppose that a and b are integers that divide the integer 'i'If a and b are relatively prime. Show…

Q: A restaurant sell features a burrito bar in which a customergets to choose one of 2 different kinds…

A: As per bartleby guidelines only one question is answered please upload the other one separately…

Q: 9. Prove that the greatest common divisor of two positive integers divides their least common…

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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.