4. n! (read "n factorial") is the product of the first n positive integers, that is 2! = 2 1, 3! = 3-2 1, etc. (a) How many zeroes are in the rightmost places (digits) of the large number 17! ? (b) Find the smallest natural number n such that n! ends in exactly 37 zeroes.

Q: find all integers n>1 such that 1^n+2^n+.....+(n-1)^n is divisible by n.

A:

Q: For a natural number n, we define n! = 1-2-3 (n − 1). n. - For example: 2! = 1-2, 3! = 1-2-3 = 6,…

A:

Q: For an integer n > 1, how many decimal stings of length n begin or end with an even digit? 5(10)"-1…

A:

Q: If a and b are positive integers, which of the following can never be the gcd(56a, 63b)? (а) 56а (b)…

A: given a and b are positive integers,  and to find gcd(56a, 63b) gcd of (56, 63) is 7  so gcd(56a,…

Q: Find all integers n such that (n^2) - 1 is prime.

A: n² - 1 = (n-1)(n+1)  Now, if (n² - 1) needs to be a prime, then, either n-1 is equal to 1 or, n+1 is…

Q: Use the contraposition to prove:  If n^2 + n + 1 is an even integer, then n is an odd integer

A: Step 1:  Step 2: Step 3: Step 4:

Q: Prove by cases For any integer n, n^2 + 5 is not divisible

A:

Q: Prove that if  n  is a positive integer, then  5 | (6^n − 1)

A:

Q: Let k be an positive integer. Prove that k(7k^2 + 5) + 6 is divisible by 6

A: Let k be an positive integer. Prove that k(7k^2 + 5) + 6 is divisible by 6

Q: prove that the sum of three odd integers is odd using indirect method. show your solution and…

A:

Q: Prove that (a) For all integers m and n, n + m and n - m are either both odd or both even.

A: Since you have posted multiple questions, according to our guidelines we can solve only first…

Q: (g) 111... 111 (126 ones) (2) Prove that the numbers given in each part cannot be prime…

A:

Q: If a and b are integers with gcd(a, b) = d Then a gcd %3D -1

A: False.

Q: Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5

A:

Q: Show that, for an integer n, if n^3 is odd, then n must be dd.

A:

Q: (a) Determine if the following numbers are prime: 101! + 78, 88! + 55 (b) Is 299 + 1 divisible by…

A: A prime number cannot be divided by any other number except 1 and itself. Now use this definition to…

Q: How many zeroes are at the end of 1000!/(500!500!)?

A: Calculate number of zeroes in 1000! and then subtract it from the number of zeroes in (500!500!).

Q: (4) For all natural numbers n, 4^n - 1 is a multiple of 3

A:

Q: True/False: Every integer greater than 1 is divisible by a prime number. True False

A:

Q: For every integer n23 I+ 2+ 2?+ +4^= 4(4n-16) %3D 3. Proof (by induction)

A:

Q: prove that Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…

Q: show that no integer of the form n^3+1 is a prime other than 2

A:

Q: how many six-digit numbers can be formed so that 2 are in the place of thousandth and 5 in the place…

A: First number can not be 0. So we can pick the first number in 9 ways. (1 to 9)

Q: Prove that if 10|n, then 10|n^2+n

A: Prove that if 10|n, then 10|n^2+n

Q: Which of the followings are true (false) in the set of real numbers? A) (Vx) (y) (x+y=10) B) (3x)…

A:

Q: Prove that n-5 is odd iff n^2-1 is odd

A: Please check step 2 for the solution.

Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

A:

Q: What is the set of O of odd positive integers less than 10?

A: The odd positive integers less than 10 are 1, 3, 5, 7, 9.

Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

A:

Q: If n is an integer greater than 1, then (a + g) - C, (a - b +2²g) + C2 (a - 2b +3² g) +(-1)" (a - nb…

A: Given n>1, a+g-C1a-b+22g+C2a-2b+32g-.....+-1na-nb+n+12gCn…