Can you think of the smallest number N such that ot can be expressed as a product of the two factors which are relatively prime in 8 ways?

Q: If you want to check that 613 is a prime number using the test for primality, what is the minimum…

A: Given number is : 613 To find what is the minimum amount of number must divide into 613 to check…

Q: Prove that the difference of squares of any two odd numbers is multiple of 8. (SOlve this task…

A: To proveDifference of squares of any two odd numbers is multiple of 8.

Q: The number of factors of prime numbers are None of the mentioned O 30 Depends on the prime number

A: No. Of factor of prime no. Means how many divisor consist of a prime no.

Q: 5. Give an example of an addition problem with three addends that would be much easier to solve…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

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: Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder 1, 5 when…

A: Let x be the required number. The number is divisible by 3, so it leaves the remainder 0 when it is…

Q: Can you use mathematical induction to show that 4 divides 32x + 7 First define the predicate P(x)

A: Yes we can use mathematical induction for Natural number x.

Q: Suppose N = Fox F₁x F2, where the three factors are pairwise prime; hence, each factor and the…

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Q: Prove that 5 does not divide 5n+3 for any integer n.

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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: Show that a number with 30 digits cannot have more than 100 prime factors.  

A: Prime factor: A factor of a number which is a prime number. For example: consider a number 26. The…

Q: 3. Use your result above to show that the number 1,111,111,113 is divisible by 3 while the number…

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Q: The number of factors of prime numberss are (None of the mentioned Depends on the prime number (2 (3

A: Prime number is only divisible by 1 and that number itself

Q: What is the smallest positive integer that has 6, 8, and 10 as factors?

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Q: a) Find the number of positive integers not exceeding 120 that are divisible by 3 or 4.

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Q: How many positive integers between 5 and 31 (a) are divisible by 3? (b) are divisible by 4? (C) are…

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Q: 1. What are the possible remainders of prime numbers greater than 2 when it is divided by 7? а. 1,…

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Q: How many integers 2 ≤ n ≤ 65 have no prime divisor less than or equal to ¹/³?

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Q: Q2: Use Fermat's method to analyze the number into its * prime factors (1745)

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Q: If the product of two integers is 2'3*5?7!" and their great- est common divisor is 2°345, what is…

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Q: how to prove √6 is irrational

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Q: Show that we can easily factor n when we know that n is the product of two primes, p and q, and we…

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Q: If n = p,P2 ... P where the p, are distinct prime numbers, how many ways can n be written as a…

A: First let us formulate the problem in a different way. Look! we have to find the total number of…

Q: The number of positive integers not exceeding 750 that are divisible by 3. 4. and 7 is: *

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Q: what two consective odd integers have a product of 63?

A: Find two odd integers.

Q: Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder 1, 5 when…

A: To find- Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder…

Q: Show that n-lis divisible by 8, if n is an odd positive integer

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Q: Which integers leave a remainder of 1 when divided by 2 and also leave a remainder of 1 when divided…

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Q: Prove that the product of an even number and an odd number is even.

A: Here, objective is to prove that the product of an even number & an odd number is even

Q: how many 6 digit numbers can be formed, if at the place of hundred thousand there are allowed two…

A: Yes your step is right.Hundredth thousand place can be chosen in 2 ways. And hundredth place can be…

Q: that if 3 divides the sum of the digits of n, then 3|n

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Q: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

Q: Find an integer that leaves a remainder of 2 whan divided by either 3 or 5, but that is divisible by…

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Q: Q2: Use Fermat's method to analyze the number into its prime factors (1745)

A: We have to find prime factorisation of 1745 using Fermat's method.

Q: Prove that 4 does not divide n2 +2 for any integer n.

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Q: 6. What is the sum of all 100 positive odd integers starting with 1? Apply your common sense if it…

A: Given n=100

Q: "For all integers n, n(n² – 1) is divisible by 6." (You may use quotient-remainder theorem and…

A: Let us consider, for all integers n a = n3 – n a = n (n2 – 1) a = n (n + 1)(n – 1)

Q: How can the sum of k * (n choose k) = n(2 to the power of n-1) be proven akgebraically ?

A: Step 1: We have to prove that, Now by the binomial theorem, Now, differentiating the both sides of…

Q: The number of primes of the form n³ – 1 is 1) 1 2) 2 3) Infinite

A: To choose correct option

Q: What is the number of integers with (a) at most n (decimal) digits; (b) exactly n digits…

A: What is the number of integers with (a) at most n (decimal) digits; (b) exactly n digits (don’t…

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: 3. There are seven students who always compare their CS test grades. They discovered that either the…

A: It is given that there are seven students who always compare their CS test grades. They discovered…

Q: composite. 10. Suppose that n = 2³ x 5² × 7 × 17³. Give the prime factoriz= (Hint: Do not work too…

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Q: 8. Show that no prime number of the form 4k +3 can divide a number of the form n2 + 1.

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Q: Find the number of positive integers ≤ 1000 that are not divisible by 5, 13, or 31?

A: .  Symbol of Greatest Integer Function  To Find: Number of positive integers ≤ 1000 that are not…

Q: Is it true that for any odd n the number 2" +1 is divisible by 3? Prove your answer.

A: We prove this by using PMI(Principle of Mathematical induction)

Q: 5 divides n -n whenever

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Q: divisible

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Q: 1.What is the smallest number that can be exactly divided by 2, 3, 4, 5  and 8 ? 2.What are the…

A: To answer the given questions.