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Number TheoryQuestion and Answers: Page 1

Question Number 218076 Answers: 0 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.

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

Question Number 217040 Answers: 2 Comments: 0

Find all positive integers n such that n divides 2^n + 1.

$Find all positive integers n such that$ $n divides 2^{n} + 1 .$

Question Number 217030 Answers: 1 Comments: 0

Find all positive integers n such that n + 1 divides n^2 + 1

$Find all positive integers n such that$ $n + 1 divides n^{2} + 1$

Question Number 216995 Answers: 3 Comments: 0

Find all prime numbers p and q such that p^2 − q^2 = 2024

$Find all prime numbers p and q$ $such that$ $p^{2} - q^{2} = 2024$

Question Number 216912 Answers: 0 Comments: 0

Find all three-digit numbers n such that 1. n is divisible by the sum of its digits. 2. n is a perfect square.

$Find all three - digit numbers n such that$ $1 . n is divisible by the sum of its digits .$ $2 . n is a perfect square .$

Question Number 216911 Answers: 1 Comments: 0

Find all positive integer x,y such that x^2 + y^2 + xy = 169

$F i n d a l l p o s i t i v e i n t e g e r x, y s u c h t h a t$ $x^{2} + y^{2} + xy = 169$

Question Number 216875 Answers: 1 Comments: 0

Let p be a prime number greater than 3. Prove that p^2 − 1 is always divisible by 24.

$Let p be a prime number greater than 3 . Prove that p^{2} - 1$ $is always divisible by 24 .$

Question Number 216783 Answers: 3 Comments: 0

Find all positive integers n such that n^2 +7n+6 is perfect square.

$F i n d a l l p o s i t i v e i n t e g e r s n s u c h t h a t$ $n^{2} + 7 n + 6 i s p e r f e c t s q u a r e .$

Question Number 216769 Answers: 1 Comments: 0

Solve for integer k,m and n: k^2 m−n^2 =8

$S o l v e f o r i n t e g e r k, m a n d n :$ $k^{2} m - n^{2} = 8$

Question Number 216664 Answers: 1 Comments: 0

Solve for non-negative integers: n^3 =3m(m+2n+1)

$Solve for non - negative integers :$ $n^{3} = 3 m (m + 2 n + 1)$

Question Number 216538 Answers: 0 Comments: 4

Find all integer solutions of 3^m =2n^2 +1. I only found m=1, 2, 5 by computer from m=1 to m=30000. Is there any greater solutions?

$Find all integer solutions of$ $3^{m} = 2 n^{2} + 1 .$ $I o n l y f o u n d m = 1, 2, 5 b y c o m p u t e r$ $f r o m m = 1 t o m = 30000 .$ $I s t h e r e a n y g r e a t e r s o l u t i o n s ?$

Question Number 215272 Answers: 1 Comments: 0

{ ((abac^(−) =(dc^(−) )^2 )),((d=((ab^(−) )/c))),((c^2 =ac^(−) )) :} abac^(−) =?

${\begin{cases} \overset{―}{a b a c} = {(\overset{―}{d c})}^{2} \\ d = \frac{\overset{―}{a b}}{c} \\ c^{2} = \overset{―}{a c} \end{cases}$ $\overset{―}{a b a c} = ?$

Question Number 214638 Answers: 1 Comments: 0

if the sum of three prime numbers is 130, what is the possible maximum of their product?

$i f t h e s u m o f t h r e e p r i m e n u m b e r s$ $i s 130, w h a t i s t h e p o s s i b l e$ $m a x i m u m o f t h e i r p r o d u c t ?$

Question Number 213756 Answers: 1 Comments: 0

Question Number 213663 Answers: 1 Comments: 0

Given a,b,c is natural numbers such that (a−b)(b−c)(c−a)=a+b+c. find min value of a+b+c

$Given a, b, c is natural numbers$ $such that (a - b) (b - c) (c - a) = a + b + c .$ $find \min value of a + b + c$

Question Number 212999 Answers: 1 Comments: 0

Find the number of non zero integer solution (x,y) to the equation ((15)/(x^2 y)) + (3/(xy)) − (2/x) = 2

$Find the number of non zero integer$ $solution (x, y) to the equation$ $\frac{15}{x^{2} y} + \frac{3}{xy} - \frac{2}{x} = 2$

Question Number 212242 Answers: 0 Comments: 7

Help

Question Number 212201 Answers: 1 Comments: 0

find the last two digits of 9^9^9 ?

$find the last two digits of 9^{9^{9}} ?$

Pg 1 Pg 2 Pg 3 Pg 4 Pg 5 Pg 6 Pg 7 Pg 8 Pg 9 Pg 10

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