Prove that if n is a square number, then n is not a perfect number.

Q: How many perfect squares are there between 1,000,000 and 9,000,000?

A:

Q: Want a rigorous proof   Prove that The set of prime numbers has no largest element.

A:

Q: Use proof by contradiction to show that, "The sum(s) of an irrational number(i) and a rational…

A:

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

A:

Q: Show that if a positive integer is odd, then it's square is odd.

A: To show that if a positive integer is odd then its square is odd

Q: 4. Show that if k is a triangular number, then 8k + 1 is a perfect square.

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: Prove that there is no rational number whose square is 12.

A:

Q: Characterize the primes p for which 63 is a quadratic residue.

A: Quadractic residue: Let p  be any odd prime number and gcd(a,p)=1. Then a is said to be a quadratic…

Q: Prove that Vp+1-\p is irrational where p is prime integer.

A:

Q: Show that if m is an integer then 3 divides m3−m.

A:

Q: What is the units' digit of the sum of the squares of the first 2019 odd positive integers?

A: To find the sum of squares of the first 2019 odd positive integers.

Q: Prove that, if a is an irrational number, then Va is also an irrational number.

A: In the given problem we have to prove that if x is an irrational number, then x3 is also an…

Q: Show that the cube for any integer is of the form 7k, or 7k ± 1

A:

Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

Q: Number theory.

A:

Q: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

Q: Are the irrational numbers closed under multiplication?

A: Use closure property.

Q: Give the primep such that 4lp +1 is a complete square:

A:

Q: a box with a square base and no top is to be made from a square piece of cardboard by cutting 4-in.…

A: Volume of a box =  Length × Width × Height

Q: 11. Prove that if n is a perfect square, then n + 2 is not a perfect square.

A:

Q: 8. We will call the integer n doubly square if n is a perfect square and its number of positive…

A:

Q: Prove that if p is a prime, then the square root of p is irrational.

A:

Q: Prove that if n is odd, then 2 is not a square in GF(5n).

A: Given n is odd number We have to show that 2 is not a square in GF5n. We know that the field GF5 is…

Q: Prove that if a is an irrational number, then a-1 is an irrational number.

A: Given:           a is an irrational number To prove:            a-1 is an irrational number

Q: If k is a positive integer, prove the k root of 6 is irrational

A: Given:Given that k is apositive integer.To find:The kth root of 6 is Irrational.

Q: Prove by contradiction that, if n is a positive integer which is not a perfect square, the sqrt(n)…

A:

Q: Prove that for all primes p2 5, p + 1 cannot be a perfect square. Recall that is a perfect square if…

A:

Q: Prove by induction that the expression for the number of diagonals in a polygon of n sides is…

A: Solution: Let Dn=nn-32 We will prove by induction that for all n∈ℕ, the total number of diagonals in…

Q: Show that square root P is irrational where P is prime by using contradiction.

A:

Q: Prove that if n is a perfect square, then n +2 is not a perfect square.

A:

Q: 12. Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle…

A:

Q: a. If p E P is a prime number, prove that the cube root of 2p is irrational.

A: We will use the basic knowledge of number theory and logic to answer this question. Note that only…

Q: А. If p 2q 5 is a prime number, prove that p2-q? is divisible by 3.

A:

Q: Prove that for any prime positive integers P,VP is an irrational number

A:

Q: 6. Prove that if x is an irrational number and Y is a nonzero rational number, then xy is…

A: to prove this we take xy is rational number then show this is contradiction.

Q: Prove or disprove: If three consecutive integers are multiplied together, and the second in order of…

A: We first look out some examples to understand the fact- choose three consecutive numbers be 1,2,3.…

Q: Give a direct proof that if m and n are even are both perfect squares, then mn is also a perfect…

A: Given that, m and n are even and both perfect squares.

Q: Let m and n be positive integers.

A: Given, m and n be positive integers.  To prove that mℤ⊕nℤ is an ideal of ℤ⊕ℤ and that…

Q: Prove that if m and n are even integers, then mn+ 1-m-n is an odd integer.

A: Given :- m and n are two even integers. To prove :- mn + 1 -m -n is odd integer

Q: Prove that if p is a prime integer, then 3√p is irrational.

A: To Prove that: If p is a prime integer, then 3√p is irrational.

Q: Show that the sum of twin primes p and p + 2 is divisible by 12, provided that p> 3.

A: Let p, p + 2 be twin primes. Then p *(p+1) + 1 = p^2 + 2p + 1 = (p+1)^2 ,  a perfect square.

Q: Show that every odd integer is the difference between two consecutive perfect squares.

A:

Q: Show that sqrt(3) is an irrational number. (And show and use that if 3 divides n^2, then 3 divides…

A:

Q: Prove that √7 is an irrational number.

A:

Q: Prove that, if z is an irrational number, then Va is also an irrational number.

A:

Q: 1. Prove that 3 is irrational number.

A:

Q: Prove that cube root of 2 is irrational by contradiction

A:

Q: 4. The area of a parallelogram is 198 square units. If the base and the height of the parallelogram…

A:

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.