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Number TheoryQuestion and Answers: Page 24

Question Number 5168 Answers: 0 Comments: 2

Find the value of 2023! (mod 2027).

$F i n d t h e v a l u e o f 2023! (m o d 2027) .$

Question Number 5165 Answers: 1 Comments: 1

what is (√(i+1)) ?

$w h a t i s$ $\sqrt{i + 1} ?$

Question Number 5158 Answers: 1 Comments: 0

why 1 + (1/2) + (1/4) + (1/8) + ........... = 2

$w h y$ $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + . . . . . . . . . . . = 2$

Question Number 5152 Answers: 0 Comments: 4

there is an ineterger a,b,c can there be an interger as a^n +b^(n+1) =c^(n+2) (n is a interger)

$t h e r e i s a n i n e t e r g e r a, b, c$ $c a n t h e r e b e a n i n t e r g e r a s$ $a^{n} + b^{n + 1} = c^{n + 2} (n i s a i n t e r g e r)$

Question Number 5144 Answers: 0 Comments: 5

Let n,j,q∈(Z^+ −{1}). Are there triples (n,j,q) such that the following conditions are satisfied altogether? (i) n=j^q (ii)n^2 =j^2 +q^2 −−−−−−−−−−−−−−−−−−−−−− Suppose then that condition (ii) above is replaced by the following condition: (iii) n^2 =rj^2 +q^2 where r∈(Z−{0,1}) What solutions (n,j,q) exist in this case?

$L e t n, j, q \in (Z^{+} - {1}) . A r e t h e r e$ $t r i p l e s (n, j, q) s u c h t h a t t h e f o l l o w i n g$ $c o n d i t i o n s a r e s a t i s f i e d a l t o g e t h e r ?$ $(i) n = j^{q}$ $(i i) n^{2} = j^{2} + q^{2}$ $- - - - - - - - - - - - - - - - - - - - - -$ $S u p p o s e t h e n t h a t c o n d i t i o n (i i) a b o v e$ $i s r e p l a c e d b y t h e f o l l o w i n g c o n d i t i o n :$ $(i i i) n^{2} = {r j}^{2} + q^{2} w h e r e r \in (Z - {0, 1})$ $W h a t s o l u t i o n s (n, j, q) e x i s t i n t h i s c a s e ?$

Question Number 5125 Answers: 2 Comments: 3

Question Number 5082 Answers: 2 Comments: 0

a^2 −b^2 =?

$a^{2} - b^{2} = ?$

Question Number 4760 Answers: 0 Comments: 2

the number 27000001 has 4 prime factors. find thier sum

$t h e n u m b e r 27000001$ $h a s 4 p r i m e f a c t o r s .$ $f i n d t h i e r s u m$

Question Number 4540 Answers: 1 Comments: 2

Let us define the positive number n with four digits a,b,c and d such that n=abcd with a,b,c,d∈Z, 1≤a≤9, 0≤b≤9, 0≤c≤9 and 0≤d≤9. Let us then say that a cool number is a four digit number, say n, such that the two digit numbers written as ab and cd are given by ab=r×s and cd=(r−1)×(s+1) for some non−negative integers r and s, r≠s. For example, 8081 has a=8,b=0 and 80=10×8= while c=8,d=1 and 81=9×9=(10−1)(8+1). So, for n=8081, r=10 while s=8. How many n, for 1000≤n≤9999, are cool? For n∈[1000,9999],n∈Z, how many n exist so that ab=r×s and cd=r×s+1? Call such n warm numbers.

$L e t u s d e f i n e t h e p o s i t i v e n u m b e r n w i t h f o u r$ $d i g i t s a, b, c a n d d s u c h t h a t n = abcd$ $w i t h a, b, c, d \in Z, 1 ⩽ a ⩽ 9, 0 ⩽ b ⩽ 9,$ $0 ⩽ c ⩽ 9 a n d 0 ⩽ d ⩽ 9 . L e t u s t h e n s a y$ $t h a t a c o o l n u m b e r i s a f o u r d i g i t n u m b e r,$ $s a y n, s u c h t h a t t h e t w o d i g i t n u m b e r s w r i t t e n a s$ $ab a n d cd a r e g i v e n b y ab = r \times s a n d$ $cd = (r - 1) \times (s + 1) f o r s o m e n o n - n e g a t i v e i n t e g e r s$ $r a n d s, r \neq s . F o r e x a m p l e, 8081 h a s$ $a = 8, b = 0 a n d 80 = 10 \times 8 = w h i l e$ $c = 8, d = 1 a n d 81 = 9 \times 9 = (10 - 1) (8 + 1) .$ $S o, f o r n = 8081, r = 10 w h i l e s = 8 .$ $H o w m a n y n, f o r 1000 ⩽ n ⩽ 9999, a r e c o o l ?$ $F o r n \in [1000, 9999], n \in Z, h o w m a n y n e x i s t$ $s o t h a t ab = r \times s a n d cd = r \times s + 1 ? C a l l$ $s u c h n w a r m n u m b e r s .$

Question Number 4362 Answers: 0 Comments: 1

Determine integers x,y,z satisfying: ax^b +by^c =cz^a a,b,c are fixed integers.

$Determine integers x, y, z satisfying :$ ${ax}^{b} + {by}^{c} = {cz}^{a}$ $a, b, c are fixed integers .$

Question Number 4358 Answers: 0 Comments: 1

Determine integer solution of bx^a +ay^b =cz^c a,b,c are fixed integers.

$Determine integer solution of$ ${bx}^{a} + {ay}^{b} = {cz}^{c}$ $a, b, c are fixed integers .$

Question Number 4242 Answers: 0 Comments: 11

Find that value of 2^2^(2∙∙∙) (continued power of 2) using analytical continuation.

$Find that value of$ $2^{2^{2 \cdot \cdot \cdot}} (continued power of 2)$ $using analytical continuation .$

Question Number 4230 Answers: 2 Comments: 1

Solve for +ve integers >0. x^2 +y^4 =z^6

$Solve for + ve integers > 0 .$ $x^{2} + y^{4} = z^{6}$

Question Number 4203 Answers: 2 Comments: 2

I have no formal background in number theory, but I′m curious of how to find positive integer solutions (x,y,z) to the equation x^n +y^n =z^n for n∈Z^− . Fermat′s last theorem led me to this. Tell me about the cases of n=−1,n=−2 and n=−3.

$I h a v e n o f o r m a l b a c k g r o u n d i n$ $n u m b e r t h e o r y, b u t I^{'} m c u r i o u s$ $o f h o w t o f i n d p o s i t i v e i n t e g e r s o l u t i o n s$ $(x, y, z) t o t h e e q u a t i o n x^{n} + y^{n} = z^{n} f o r$ $n \in Z^{-} . {F e r m a t}^{'} s l a s t t h e o r e m l e d$ $m e t o t h i s . T e l l m e a b o u t t h e c a s e s$ $o f n = - 1, n = - 2 a n d n = - 3 .$

Question Number 4199 Answers: 1 Comments: 0

Let p∈P and m∈Z^+ . Find (p,m) such that p^(m−1) (p−1)=146410.

$L e t p \in P a n d m \in Z^{+} .$ $F i n d (p, m) s u c h t h a t p^{m - 1} (p - 1) = 146410 .$

Question Number 3685 Answers: 1 Comments: 0

let p_i be the i^(th) prime Does the fillowing sum converge: Σ_(i=1) ^∞ (p_i /p_(i+1) ) (p_1 =2, p_2 =3, p_3 =5, ...)

$let p_{i} be the i^{th} prime$ $Does the fillowing sum converge :$ $\underset{i = 1}{\sum^{\infty}} \frac{p_{i}}{p_{i + 1}}$ $(p_{1} = 2, p_{2} = 3, p_{3} = 5, . . .)$

Question Number 3659 Answers: 0 Comments: 6

p_i = ith prime let: ρ=p_n −p_(n−1) is: lim_(n→∞) ρ=∞?

$p_{i} = i th prime$ $let : ρ = p_{n} - p_{n - 1}$ $is :$ $\lim_{n \to \infty} ρ = \infty ?$

Question Number 3630 Answers: 0 Comments: 0

Prove that sum of all prime numbers p such that n≤p≤2^k n is ≥2^k n. Σ_(n≤p≤2^k n) p ≥ 2^k n (p−prime)

$Prove that sum of all prime numbers p$ $such that n ⩽ p ⩽ 2^{k} n is ⩾ 2^{k} n .$ $\sum_{n ⩽ p ⩽ 2^{k} n} p ⩾ 2^{k} n (p - p r i m e)$

Question Number 3644 Answers: 1 Comments: 1

An hexagon of unit side is drawn on plane. Draw a square having the same area as the hexagon using only unmarked ruler and compass. What if an n−gon with unit edges is given? Is it always possible to draw a square of the same area as n−gon using ruler and compass.

$An hexagon of unit side is drawn on plane .$ $Draw a square having the same area as the$ $hexagon using only unmarked ruler and$ $compass .$ $What if an n - gon with unit edges is given ?$ $Is it always possible to draw a square$ $of the same area as n - gon using ruler$ $and compass .$

Question Number 3595 Answers: 0 Comments: 10

I just thought of something I am curious in figuring out. All integer numbers can be made up by prime factors. That is: n=p_1 ×p_2 ×...×p_i n∈Z p_k ∈P Are there an inifinite number of numbers that are the sum of prime numbers? That is: P=p_1 +p_2 +...+p_i P,p_k ∈P For example: 2+3=5 2+3+3+5=13 etc. Are all of these special primes odd? What else can we work out?

$I just thought of something I am curious$ $in figuring out .$ $All integer numbers can be made up by$ $p r i m e f a c t o r s . That is :$ $n = p_{1} \times p_{2} \times . . . \times p_{i}$ $n \in Z p_{k} \in P$ $Are there an inifinite number of numbers$ $that are the s u m of p r i m e n u m b e r s ?$ $That is :$ $P = p_{1} + p_{2} + . . . + p_{i}$ $P, p_{k} \in P$ $For example :$ $2 + 3 = 5$ $2 + 3 + 3 + 5 = 13$ $e t c .$ $Are all of these special primes odd ?$ $What else can we work out ?$

Question Number 3519 Answers: 1 Comments: 5

n is a number such that regular n−gon is possible with straightedge and compass only. ∗Write first thirty values of n. ∗What are other properties of such numbers ? ∗If values of n are arranged in order, what is the formula for generating Nth number?

$n i s a n u m b e r s u c h t h a t r e g u l a r n - g o n i s$ $p o s s i b l e w i t h s t r a i g h t e d g e a n d c o m p a s s o n l y .$ $* W r i t e f i r s t t h i r t y v a l u e s o f n .$ $* W h a t a r e o t h e r p r o p e r t i e s o f s u c h n u m b e r s ?$ $* I f v a l u e s o f n a r e a r r a n g e d i n o r d e r, w h a t i s$ $t h e f o r m u l a f o r g e n e r a t i n g N t h n u m b e r ?$

Question Number 3296 Answers: 1 Comments: 3

If n,p,q∈Z^+ and p and q are coprimes, then prove that HCF of (n^p −1) and (n^q −1) is (n−1). Assume n>1.

$If n, p, q \in Z^{+} and p and q are coprimes,$ $then prove that HCF of (n^{p} - 1) and (n^{q} - 1) is (n - 1) .$ $Assume n > 1 .$

Question Number 3205 Answers: 1 Comments: 9

Suggest minimum number of weights ,two peices of each, to weigh upto at least 60 kg(in whole kg′s) in a common balance.

$S u g g e s t m i n i m u m n u m b e r o f w e i g h t s, t w o p e i c e s o f e a c h,$ $t o w e i g h u p t o a t l e a s t 60 k g (i n w h o l e {k g}^{'} s) i n a c o m m o n$ $b a l a n c e .$

Question Number 3172 Answers: 1 Comments: 1

How many different clock−type dials can be made containing first n natual numbers with the property that sum of any two numbers of consecutive positions be a prime number. N={1,2,3,...}

$H o w m a n y d i f f e r e n t c l o c k - t y p e d i a l s c a n b e m a d e$ $c o n t a i n i n g f i r s t n n a t u a l n u m b e r s w i t h t h e p r o p e r t y$ $t h a t s u m o f a n y t w o n u m b e r s o f c o n s e c u t i v e p o s i t i o n s b e$ $a p r i m e n u m b e r .$ $N = {1, 2, 3, . . .}$

Question Number 3146 Answers: 1 Comments: 0

Change the order of numbers on a clock−dial so that sum of any two numbers of consecutive positions may be prime.

$C h a n g e t h e o r d e r o f n u m b e r s o n a c l o c k - d i a l s o t h a t$ $s u m o f a n y t w o n u m b e r s o f c o n s e c u t i v e p o s i t i o n s m a y$ $b e p r i m e .$

Question Number 2878 Answers: 2 Comments: 1

How many ordered pairs (m,n) are there for 1≤m,n≤100 such that 7^m +7^n is divisble by 5?

$H o w m a n y o r d e r e d p a i r s (m, n) a r e t h e r e$ $f o r 1 ⩽ m, n ⩽ 100 s u c h t h a t 7^{m} + 7^{n}$ $i s d i v i s b l e b y 5 ?$

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