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Number TheoryQuestion and Answers: Page 1

Question Number 218208 Answers: 1 Comments: 1

This question is really important Prove or disprove that lim_(n→∞) ((3^n m+3^(n−1) )/2^(⌈(n/2)⌉) ) + (3^(n−1) /2^n ) the limit exists for m ∈ N \B where B = {n ∣ log_2 (n) ∈ N }

$T h i s q u e s t i o n i s r e a l l y i m p o r t a n t$ $P r o v e o r d i s p r o v e t h a t$ $\lim_{n \to \infty} \frac{3^{n} m + 3^{n - 1}}{2^{⌈ \frac{n}{2} ⌉}} + \frac{3^{n - 1}}{2^{n}}$ $t h e l i m i t e x i s t s f o r m \in N ∖ B$ $w h e r e B = {n ∣ {l o g}_{2} (n) \in N}$

Question Number 218076 Answers: 2 Comments: 0

How many ways to arrnge the letters ABCCCDEFG (1) in general . (2) all 3 Cs must be together (3) only 2 Cs must be together (4) no 2 or 3 Cs be together (5) no letter still in its original place .

$H o w m a n y w a y s t o a r r n g e$ $t h e l e t t e r s A B C C C D E F G$ $(1) i n g e n e r a l .$ $(2) a l l 3 C s m u s t b e t o g e t h e r$ $(3) o n l y 2 C s m u s t b e t o g e t h e r$ $(4) n o 2 o r 3 C s b e t o g e t h e r$ $(5) n o l e t t e r s t i l l i n i t s$ $o r i g i n a l p l a c e .$

Question Number 217958 Answers: 2 Comments: 2

a,b,c∈Z^+ and a^2 +b^2 +c^2 +ab+bc+ca=2025 find out all triplets (a,b,c).

$a, b, c \in Z^{+} a n d$ $a^{2} + b^{2} + c^{2} + a b + b c + c a = 2025$ $f i n d o u t a l l t r i p l e t s (a, b, c) .$

Question Number 217952 Answers: 2 Comments: 0

If x∈Z ∧y non-negative integer such that x^2 +10x+23=2^y find out x,y

$I f x \in Z \land y n o n - n e g a t i v e i n t e g e r$ $s u c h t h a t$ $x^{2} + 10 x + 23 = 2^{y}$ $f i n d o u t x, y$

Question Number 217244 Answers: 1 Comments: 0

Find all two-digit numbers such that when the number is divided by the sum of its digits the quotient is 4 and the remainder is 3.

$Find all two - digit numbers such that$ $when the number is divided by$ $the sum of its digits the quotient$ $is 4 and the remainder is 3 .$

Question Number 217132 Answers: 0 Comments: 0

Find all integers n> 1 such that n divides 2^(n−1) + 3^(n−1) .

$Find all integers n > 1 such that$ $n divides 2^{n - 1} + 3^{n - 1} .$

Question Number 217130 Answers: 0 Comments: 0

Prove that for every integer n≥2 the number n^4 + 4^n is composite.

$Prove that for every integer n ⩾ 2 the number n^{4} + 4^{n} is$ $c o mposite .$

Question Number 217129 Answers: 1 Comments: 2

prove that if an integer n is not divisible by 2 or 3 then n^2 ≡1(mod 24)

$p r o v e t h a t i f a n i n t e g e r n i s n o t d i v i s i b l e b y 2 o r 3$ $t h e n n^{2} \equiv 1 (m o d 24)$

Question Number 217066 Answers: 1 Comments: 0

Find all integer x,y such that x^2 −y^2 =100

$F i n d a l l i n t e g e r x, y s u c h t h a t$ $x^{2} - y^{2} = 100$

Question Number 217071 Answers: 1 Comments: 0

Find all two-digit numbers that are equal to four times the sum of their digits. Solve this using at least two different methods and verify your answers.

$Find all two - digit numbers that are equal to four times the sum$ $of their digits . Solve this using at least two different methods$ $and verify your answers .$