QUESTION 8 Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder 1, 5 when divided by 4 and 7 respectively? (Show your work)

Q: Euclid of Alexandria is credited with writing a series of 13 mathematics books called The Elements…

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Q: How many 3-digit positive integers exist that when divided by 7 leave a remainder of 57

A: 3 digit positive integers start from 100 and ends with 999. Least positive 3 digit integer which…

Q: Question 32 Given a universal set containing prime numbers between -20 and 20 and composite numbers…

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Q: A natural number with 5 as the units digit is given. When the units digit is erased, the obtained…

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Q: Question 9 What are the prime factors of 30?

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Q: Serial numbers for a product are to be made using 2 letters followed by 2 digits. The letters are to…

A: Number of possibilities: Here, the password consisting of 2 letters followed by 2 digits. There are…

Q: n2 1, n - nis divisible by 3.

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Q: Question: How can you show 5+7+ 9 +...+ (2n + 3) = n (n + 4) is true for all natural numbers using…

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Q: The numbers 6, 14, and 15 are factors of the natural number N.  What is the smallest value that N…

A: Given that the numbers 6, 14 and 15 are factors of the natural number N To find: the smallest value…

Q: A sequence of positive integers starts with one 1, followed by two 2s, three 3s, and so on. (Each…

A: Given:     The sequence is:     1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,... To get this number, we have to…

Q: Serial numbers for a product are to be made using 2 letters followed by 3 digits. The letters are to…

A: Consider the serial number with no repeats is, No repeats 9 ways 8 ways 1st letter 2nd letter…

Q: Question 9 ( What can we do to show that 313 is prime? We can determine whether the smallest prime…

A: Prove that 313 is a prime number.

Q: Question 9 What is smallest integer that is greater than -10.5? Question 5 What is largest integer…

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Q: When Carl Friedrich Gauss (1777 - 1855) was at school his teacher wanted time off and gave the class…

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Q: Allowing digits and special characters (as well as case-sensitive letters) creates an enormous…

A: As per the policy we are allowed to do only 1 question so I attempted question 3. Please re-upload…

Q: QUESTION 1 Find an integer x between 220 and 260 that satisfies x = 39^123 (mod 41). Your answer…

A: We need to find integer x between 220 and 260 that satisfies x = 39123 mod 41 Clearly , 39 and 41…

Q: Problem 2. Determine whether 2^n − 1 is prime for each of the primes not exceeding 100

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Q: How many four-digit combinations can be formed using digits 0 to 9?

A: Total number of digits n=10 Number of digits for combination r=4

Q: Identify an expression that is divisible by 5 from the following expressions.   1.  5^h−2^h where…

A: 1. 5h-2h where h is integer. terms are 51-21,52-22,53-23,.....3,21,117.... ⇒  Not divisible by 5. 2.…

Q: Question 6. We can use (C) to represent C(n, k). Show that (") is divisible by n if n is a prime…

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Q: Question 1 Find all integers n>1 such that 1" + 2"+...+(n – 1)" is divisible by n.

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Q: Verify the expression 5^m−2^m is divisible by     1. 3 whenever m is a negative integer   2. 3…

A: Just plug in positive and negative values of m, and you will get an idea that given expression is…

Q: Question 2 Given two integers n and m such that their remainders after division by 5 equal to 1.…

A: Given that n and m be two integers such that remainder after division by 5 is 1. Then there exists…

Q: Question 3 3! x (x - 1 =0),where domain consists of all integer numbers True False A Moving to…

A: this is true because we can write this in the form x^2-x=0 so this becomes a quadratic equations

Q: Problem 2: Prove that for each n E No, - I« (2n)! (n + k) = (n + 1)(n +2) - - · (2n) n! k-1 is…

A: It  is prove by principle of mathematical induction P(n): (2n)!n! is divisible by 2n n=1P(1) : 2!1!…

Q: 5. Assume n > 3 is an integer. n!(n +1)! (n – 1)!(n +2)! (a) Simplify

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Q: The X numbers may be defined recursively by X, = 0,X, = 2 and X, = (X,-2)² + 2X,n-1 a) Please find…

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Q: Serial numbers for a product are to be made using 2 letters followed by 3 digits. The letters are to…

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Q: How can you use Pascal's triangle to expand (2x - 3y)? Drag an expression or value into each box to…

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Q: QUESTION 3 Arrange steps in approaching proof of the theorem. If direct proof doesn't lead to any…

A: Note: “Since you have asked multiple question, we will solve the first question for you. If you want…

Q: A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes.…

A: A number n! written in base 6 and 8 notation, where its base 6 representation ends with 10 zeros and…

Q: How can you use Pascal's triangle to expand (2x - 3y2)"? Drag an expression or value into each box…

A: We can answer the question as below

Q: How many integers from 1 to 1000 are divisible by at least one of  the numbers 2, 3, 5?

A: Considering integers from 1 to 1000, Total integers  = 1000 Let n(A) represent the integers…

Q: Problem B Show that 3. 4" + 51 is divisible by 3 and 9 for all positive integers n.

A: We need to show that 3·4n+51 is divisible by 3  by using the principle of mathematical induction…

Q: It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a…

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Q: Question 5 Use Pascal's triangle to expand your binomial expression. A (1 + x)³ B (1-x)³ C (1 + x)4…

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Q: Question 3: If n is a positive natural number (like 1, 2, 3 etc.), what number would belong in the…

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Q: medium second and another arrangement is medium first, large second. How many such arrangement are…

A: Hezekiah has six boxes of six different sizes. Four box are taken at a time and arrange in definite…

Q: A restaurant offers a dinner special that lets you choose from 10 entrees, 8 side dishes, and 13…

A: Multiplication rule of counting:If a task consists of a sequence of choices in which there are ‘a’…

Q: Question 27 This Test: 40 pts possible If of all rabbits in a country are in one state and of the…

A: Let total number of rabbit = 7 × 32 = 224

Q: QUESTION 2 For a GP, 1/2, 1/4, 1/8, … the sum of the first term S1 = 1/2, the sum of the first two…

A: We'll answer one question since the exact one wasn't specified. Please submit a new question…

Q: 10. Find the 8th term in the 13th row of Pascal's triangle. Show your work here:

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Q: 10. A small population of 27 birds on an island are marked so that each individual can be…

A: Combination

Q: In how many ways can the letters be arranged so that all the vowels come together? Word is…

A: Given : In how many ways can the letters be arranged so that all the vowels come together? Word is…

Q: A population of fruit flies is growing in such a way that each generation is 1.15 times as large as…

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Q: If n> 5 is an integer, prove the identity () - OO(;)-O0C:")- O0(:)--(":)OO +..+ by counting in two…

A: We are given a formal containing sum of binomial coefficients and we have to prove it using two…

Q: Question 2) Using back substitution method find all the integer x of the following system (Chinese…

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Q: Question 8 of 13, Step 1 of 1 7/13 Correct A positive integer is 17 more than 13 times another.…

A: The solution is given below:

Q: a)Select a variety of one- and two digit numbers between 1 and 99 and multiply by 9. Record your…

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