11. If p is a prime and p divides 4.":,, thenp divides " for some i.

Q: 1. If p>9 2 5 and p and q are both primes, prove that 24 p - q.

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Q: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

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Q: what

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Q: 2. Show that In (- - ) = ( + 2nn)i where n = | .... .... ..

A: Consider the following expression: x=ln-12-32iex=-12-32icosx+isinx=-12-32i

Q: Let A be the set of positive integers that are less than or equal to 40 and are multiples of 4 and…

A: Given: Let A be the set of positive integers that are less than or equal to 40 and are multiples of…

Q: Prove: in L number psiz 12, for every prime poiz, [p] = [1], [5) , [7) , or [11]

A:

Q: 1. Will there be at least two people who celebrate their birthday on the same day Yes, definitely O…

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

Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

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Q: Let n be an integer with n ≥ 2. Suppose that for every prime               p ≤√n, p does not divide…

A: given,                                                  p≤n p doesn't divide n. let n is not a prime…

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

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: 7. Prove: Vn,meZ, if n and m are odd, then n·m is odd.

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Q: Let X~ N (24, 49), Find P( 10<X< 38)

A: Given sample are: 24, 49 Mean of these samples are: μ=24+492=36.5 Standard deviation:…

Q: 23. Find de V100- tan O coton 2 Opbon 1

A: We have the given integral as ∫dx100-9x2 On solving the given integral furthermore, we get the…

Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

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Q: Given that it will be Friday in 2 days, which of the following statement is TRUE? - It will be…

A: Given : It will be Friday in two days  Today is Wednesday

Q: Show that if p > 1 and p divides (p − 1)! + 1, then p is prime.

A: Proof by contradiction: Assume p is not prime ⇒p is composite

Q: We need to attempt to find an exact value of n such that  P(r ≥ 6) = 0.98. Clearly to guarantee…

A: We have to fill blanks from given statement Clearly to guarantee at least six people are literate,…

Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p ≥ 5, at least one of…

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Q: () - (" ;')-(,".) Show that: n+1

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Q: Prove bey wnduc hiou that for n22.

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Q: If p>=q>=5 and p and are both primes, prove that 24| p^2-q^2.

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

A: See the handwritten solution

Q: Given any odd prime p, prove that: 2(p−3)! ≡−1 mod p. (Hint: Start with 2(p−3)!≡x mod p. Then work…

A: Let p be any odd prime number.  So we start with  2p-3!≡x mod p Also we have p-2≡-2 mod pp-1≡-1 mod…

Q: Suppose that m, n are two integers such that m 2n2 = 1. Does it follow that m and n are relatively…

A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

Q: If a and b are relatively prime then alb

A: Given below the detailed solution

Q: Given 51 distinct positive integers Strictly of Prove two trom to Sum 99 less than 100,

A: Please find the answer in next step

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: Prove VEN+, 5+ 3n ≥ 22n+1

A: Introduction: The idea of mathematical induction aids in the proof of mathematical conclusions and…

Q: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

A: The given term is n2+5n

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: Prove the fallowing egudion for every n,2 by hathematical enduelion. 35.6+35 6+ 35.63+ -+35.6"= 7.6"

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Q: prove by contrapositive : for all n,m integers if nm is even then n is even or m is even

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Q: of the apples is equal to 3 of the oranges. If there are 12 mor 5 In a fruit stall, - - annles than…

A: Fraction:  A small part of something.  In other words, numerical value that is not a whole number A…

Q: If gcd(m,p) = 1 and p is prime, show that gcd(m,pk ) = 1 for all k≥ 1.

A: To show, If gcd(m, p)=1 and p is prime, show that gcd(m, pk) = 1 for all k≥1.

Q: . How many different values are there for 4" (mod 7), as n ranges over all positive integers?

A: Note - As per our guidelines we have to answer first question if multiple questions are posted. So…

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 +q2 + r2.

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Q: 11. Establish the following relation between Fibonacci and Lucas numbers: L, = un + 2un-1 for all n…

A: Remark: We know that both the Fibonacci numbers and  Lucas numbers are obtained as the sum of its…

Q: Prove that 21 divides 2n-) 4nt whenever is Posi tive in teger in

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Q: 2. If p > q > 5 and P and q are both primes, prove that 24 p - q² |

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Q: 4. Find x in ASTU. * 17 15 O 2 O 8 O v32 O V514

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Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

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Q: Let p and q be prime numbers. (1) If p is not equal to q, how many generators does Zpq have? (2) If…

A: Given p and q be primes.

Q: if p> q > 5 and p, q are both primes, prove that 24 | p² – q

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Q: Prove that if n > 1 is a number with φ(n) = n/k, then k = 2,3.

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Q: Since the next prime after 7 is​ 11, the smallest composite number whose prime factors are all…

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Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 + q2 + r2.

A: We have to prove that 3 divides p2+q2+r2, where p,q and r primes other than 3.

Q: Prove that If p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

A:

Q: For n > 3, prove that the numbers ±5, ±52, · reduced residue system modulo 2". form a

A: A subset R of the integers is called reduced a residue system modulo n if (i) gcdr,n =1 for each r…

Q: 2. Show directly that sup(4) = 1 for A = {1- : n eN}.

A: As we know that, 1&gt;1-1n ∀ n∈ℕ Therefore 1 is the upper bound of the set 1-1n:n∈ℕ. And for each…

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.