Find the total number of distinct prime divisors of the number n. Identify the greatest prime divisor. 1. n=( 51!)(67!)+(78!) 2. n= ((50!/30!)-25!)

Q: Let p>5 be the prime number. Prove that the expression p4-10p2+9 is divisible by 1920.

A: Let, p>5 be the prime number. Prove that the expression p4-10p2+9 is divisible by 1920.

Q: Show that in a sequence of m integers there exists one or more consecutive terms with a sum…

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Q: Find the greatest natural number n such that n + 10 divides nº + 100.

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Q: Use mathematical induction to show that the sum of the cubes of three consecutive integers is…

A: To Use mathematical induction to show that the sum of the cubes of three consecutive integers is…

Q: Number of integers n with 4 prime factorizations and divisible by 5, where n less than or equal 2021…

A: we shall consider all the integers less than or equal to 2021, and divisible by 5. And also having 4…

Q: (a) In how many ways can we select 4 people from a group of 10 people and award them first through…

A: a) Since we want to give them awards by ranking, so here arrangement matters. We will have to use…

Q: 3. Let n be an integer greater than 2. Explain why n! +2, n! +3, n! +4, n! + (n – 1), n! +n are all…

A: We have to explain that n!+2, n!+3,…,n!+n-1,n!+n are all composite numbers.

Q: Find how many positive integers with exactly four decimal digits, that is, positive integers between…

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Q: 24. Show that if p is a prime and k is an integer such that 1sksp-1. then p divides ().

A: 24. we have to show  that for a prime p with r≤k≤p-1      p divides  pk  Notation: pk=p!p-k! k!

Q: 2. Determine the highest power of 5 that divides each of the following positive integers. a) 112,250…

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Q: The integers 77 and 105 are relatively prime. صواب İhi

A: By finding the factors of given numbers you can find the solution :

Q: 4. Find the largest value of n such that 2"|16!.

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Q: 5. Let n be a positive integer and p1,P2,P3, . „Pn be n distinct primes. If m is a positive integer…

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Q: Suppose a,b,n are integers and n>0 s.t. 63a^5b^4=3575n^3, what is the smallest possible n. Explain…

A: The objective is to find the smallest value of n such that 63a5b4=3575n3.

Q: a) In how many ways can the letters in the word SUCCESS be arranged? b) In how many ways can the…

A: a) There is a total of 7 letters in the word SUCCESS. Among them 3 are S and 2 are U. So the total…

Q: 16. What is the greatest integer N such that (10 – N) + (16 - N)> 5?

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Q: How many elements are there in the greatest conjugacy class of 5 ?S

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Q: divisors

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Q: The prime factorization of 7007 is 7.113.13 O 73.11.13 7.11.13 72.11.13

A: We need to find prime factorization of 7007.

Q: Find an integer between 100 and 150 divisible by 9.   How do i figure this out, I am completely…

A: A number is divisible by 9 if the sum of the digits is divisible by 9.

Q: Determine the sum of all the two-digit multiples of 5.

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Q: 6.) Find the sum of the first 50 positive integers ending in 6.

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Q: B. Using induction, prove that 8" -1 is divisible by 7, where ne N

A: (B) 8n-1 is divisible by 7 Now for n=1 81-17=1 For n=2 82-17=64-17=637=9 Now, assume that 8n-1 is…

Q: 6. How many integers from 1 through 100 must you pick in order to be sure of getting one that is…

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Q: Find the least positive integer n for which 2" + 5" - n is a multiple of 1000.

A: To find the value of n substitute n=1,2,3... multiple of 1000 is a certain number divided by 1000…

Q: How many integers are greater than 120, less than 540 and are divisible by 5 and 3? 26 28 None of…

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Q: Find the which value of n for where vin)' is the divisors ofn Smallest vin) =6 number of positive

A: The given problem is related to number theory. Here, we have to find the smallest value of n for…

Q: hree young brother-sister pairs from different families need to take a trip in a van. These six…

A: Given: n=6 Three young brother-sister pairs  Consider a is the first pair, b is second and c is the…

Q: How many positive integers greater than 5 and less than 2020 are divisible by 7? are divisible by…

A: 1) Here , 2020/7 = 288.57 So 288 multiples of 7 are there between 5 to 2020 So , 288 integers are…

Q: = 55. Determine the value of n if (5)

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Q: What is the greatest common divisor of 24 and 44? n If n is a natural number such that gcd(800, n) =…

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Q: 5. Find the number of positive integers less than 700 that are divisible by 3 but not divisible by…

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Q: Let p>5 be the prime number.Prove that the expression p4-10p2+9 is divisible by 1920.

A: Given: Let p>5 be the prime number

Q: (b) Give two different interpretations of the bino- mial coefficient (") for non-negative integers n…

A: First we will give two different interpretations of the binomial coefficient nk for non-negative…

Q: 6. (a) Show, using the pigeonhole principle, that in any set of 5 integers, at least two have the…

A: (a) Show that any set of 5 integers at least have two the same remainder when it is divided by 4…

Q: 2. The Hellenistic mathematician Nicomachus studied the following concept related to perfect…

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Q: The prime factorization of 7007 is 7.11.13 72.11.13 73.11.13 7.11³.13 o

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Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

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Q: Show that 5 | n5-n for all integers n.

A: To prove the above statement we will use the principle of mathematical induction which is divided in…

Q: find the sum of all integers between 300 and 800 that are exactly divisible by 14

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Q: What is the largest n such that 16 and 25 are in the same remainder class modulo n? Write out the…

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Q: Find all values of primes p for which (¹¹) = 1 Р

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Q: Show by induction that 21" – 8" is divisible by 13 for all n> 0

A: For the solution follow the next steps.

Q: se mathematical induction to show    n ≥ 0, 7n - 1 is divisible by 6

A: Given: n≥0, 7n-1 To prove: 7n - 1 is divisible by 6

Q: Show by induction that 19" – 8" is divisible by 11 for all n >0

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Q: IV. What are the greatest common divisors of these pairs of integers? Show your solutions. 13. 33…

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Q: Find the sum of the divisors of 120. Generalise the result for an arbitrary natural number M

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Q: 4. Find them if: (a) lcm of two consecutive positive integers is 812; (b) lcm of twin primes is 899.

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Q: Show that...

A: To show: That n0=nn for n≥0. Concept used: The combination are shown below: Crn=n!r!n-r!…