### Number Theory

Find all the integers such that when the final digit is deleted the new integer divides the original one. Can you generalize this to deleting other digits?

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Find all the integers such that when the final digit is deleted the new integer divides the original one. Can you generalize this to deleting other digits?

Decipher the following CFQGE KAZEMF ZMAGVMC NMO VYSV which was obtained by a formula of the type y=kx (mod 26)

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Prove that if p is a prime number, then p divides , for all n≥p. Here [r] denotes the greatest integer ≤ r , for any real number r. Does this result generalize to a result about instead of p ?

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For a mapping %:A -> B, let == denote the kernel equivalence of %, and let *:A -> A== denote the natural mapping. Define $:A== -> B by $([a]) = %(a) for every equivalence class [a] in A==. 1. Show that $ is well defined and one-to-one, and that $ is onto if % is onto. Furthermore, show that % = $*, so that % is the composite

Given ( INTEGRAL ln square(x)dx, as x from n to n+1 ) = ( INTEGRAL ln square (n+x)dx, as x from 0 to 1 ) = ( INTEGRAL [[ln(n+x) - ln(x) + ln(n)]square] dx, as x from 0 to 1 ), (a) Verify that ( LIMIT (n/ln(n)) [INTEGRAL (ln square (x)dx) - (ln square (n))] as n approach to the infinity ) = 1 (b) Compute LIMIT ((n square)/

Here is what the problem asks for: Give an example of a polynomial function f of degree 5 such that the only real roots of f(x) are -2,1,6 and f(2)=32. Show that your example works and leave f(x) in factored form.

Please see the attached file for the fully formatted problems. Let g be a function which can be differentiated four times on the interval [-1,1]. Denote . 1) Show that when g is a polynomial of degree less than or equal to 3. 2) Let P be the interpolation polynomial of f at the points -1, , , 1. a) Show that . b

Let z be a complex number z=x+iy x <>0 and y<>0 Prove: 1. If z+1/z is real then |z|=1 2. If |z|=1 then z+1/z is real

Polynomial with real coefficient

When the accountants for lose-a-digit Computer, Inc. had finished preparing their annual budget, they presented the final figures to the president, I.M. Smart. "It looks like a good year," he exclaimed. "The amount of the budget just happens to be the smallest number of cents (other than one cent) that is a perfect square, a per

Simplify: (a/(a-b))+(b/a^2+ab+b^2))- (2/(a^3-b^3)

(a) Let G = GL(2,R) be the general linear group. Let H=GL(2,Q) and K= SL(2,R) = {A is an element of G: det (A) =1} Show that H,K are subgroups of G (b) Let p be a prime number and a is an element of Z. Prove that a^p ,is equivalent to, a mod p

Prove the following conjectures, that for all N in the set of Natural Numbers... Question continued in attachment.

What is (6^6)+(6^5) in (mod5)?

What is the ones digit of the number (23)^23?

The natural number 28A9B consists of different numbers and A is not equal to 0. When the number is divided by 9 the remainder is 7 and when it is divided by 5, the remainder is 1. What is A-B=?

What is the result of ((-1/2)^-3)^2?

What is the result of 0.1/0.01 + 0.01/0.001 + 0.001/0.0001=?

Show that the square root of a prime number is not rational.