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Number TheoryQuestion and Answers: Page 25

Question Number 2842 Answers: 0 Comments: 0

Prove that π=3.14... is irrational.

$P r o v e t h a t π = 3.14 . . . i s i r r a t i o n a l .$

Question Number 2820 Answers: 3 Comments: 11

We have the idea of Phythagorian triples as solutions (x,y,z) to the equation x^2 +y^2 =z^2 where x,y,z∈Z^+ . How frequently do solutions (x,y,z,t) to the equation x^2 +y^2 +z^2 =t^2 arise for x,y,z,t being integers between 1 and 100 inclusively? What solutions exist for t=12 and x,y,z∈Z^+ ?

$W e h a v e t h e i d e a o f P h y t h a g o r i a n t r i p l e s$ $a s s o l u t i o n s (x, y, z) t o t h e e q u a t i o n$ $x^{2} + y^{2} = z^{2}$ $w h e r e x, y, z \in Z^{+} .$ $H o w f r e q u e n t l y d o s o l u t i o n s (x, y, z, t) t o t h e e q u a t i o n$ $x^{2} + y^{2} + z^{2} = t^{2}$ $a r i s e f o r x, y, z, t b e i n g i n t e g e r s b e t w e e n 1 a n d 100 i n c l u s i v e l y ?$ $W h a t s o l u t i o n s e x i s t f o r t = 12 a n d x, y, z \in Z^{+} ?$

Question Number 2815 Answers: 1 Comments: 3

η(s)=Σ_(n=1) ^∞ (((−1)^(n−1) )/n^s ) Dirichlet eta function prove that η(s)=(1−2^(1−s) )ζ(s)

$η (s) = \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n - 1}}{n^{s}} Dirichlet eta function$ $prove that$ $η (s) = (1 - 2^{1 - s}) ζ (s)$

Question Number 2699 Answers: 1 Comments: 0

Find the remainder when 3^(215) is divided by 43.

$Find the remainder when$ $3^{215} is divided by 43 .$

Question Number 2603 Answers: 2 Comments: 0

You have a 3 litre jug and a 5 litre jug. Make 4 litres.

$Y o u h a v e a 3 l i t r e j u g a n d a 5 l i t r e j u g . M a k e 4 l i t r e s .$

Question Number 2575 Answers: 2 Comments: 4

a_1 =0 a_n =27×a_(n−1) +(n−1) Σ_(k=1) ^m a_k =?

$a_{1} = 0$ $a_{n} = 27 \times a_{n - 1} + (n - 1)$ $\underset{k = 1}{\sum^{m}} a_{k} = ?$

Question Number 2432 Answers: 1 Comments: 0

What is the sum of digits of 3333^(4444) , Say sum of all digits of 3333^(4444) is A, If A>10 then sum all digits of A. This process is repeated until a single digits sum x in obtained. x=?

$What is the sum of digits of 3333^{4444},$ $Say sum of all digits of 3333^{4444} is A,$ $If A > 10 then sum all digits of A .$ $This process is repeated until a single$ $digits sum x in obtained .$ $x = ?$

Question Number 2387 Answers: 0 Comments: 4

How many 0s at the end of 1000!? What is the first non zero digits from the right?

$How many 0 s at the end of 1000! ?$ $What is the first non zero digits from the right ?$

Question Number 2381 Answers: 1 Comments: 2

Of the numbers 1, 2, 3, ... , 6000, how many are not multiples of 2, 3 or 5?

$O f t h e n u m b e r s 1, 2, 3, . . ., 6000,$ $h o w m a n y a r e n o t m u l t i p l e s o f 2, 3 o r 5 ?$

Question Number 2148 Answers: 1 Comments: 0

Is 3^(2015) −2^(2015) prime?

$I s 3^{2015} - 2^{2015} p r i m e ?$

Question Number 1942 Answers: 1 Comments: 1

Let N be a positive integer with prime factorisation N=p_1 ^m_1 p_2 ^m_2 p_3 ^m_3 ×...×p_(n−1) ^m_(n−1) p_n ^m_n where n,m_i ∈Z^+ and p_r is prime. How many proper factors does N have? Investigate cases where n=1,n=2, n=3 and n=4. What is the smallest positive integer with 12 proper factors? What is the smallest positive integer with at least 12 proper factors? (A proper factor of a positive number N is positive nteger M such that M≠1 and M≠N.)

$L e t N b e a p o s i t i v e i n t e g e r w i t h p r i m e$ $f a c t o r i s a t i o n$ $N = p_{1}^{m_{1}} p_{2}^{m_{2}} p_{3}^{m_{3}} \times . . . \times p_{n - 1}^{m_{n - 1}} p_{n}^{m_{n}}$ $w h e r e n, m_{i} \in Z^{+} a n d p_{r} i s p r i m e .$ $H o w m a n y p r o p e r f a c t o r s d o e s N h a v e ?$ $I n v e s t i g a t e c a s e s w h e r e n = 1, n = 2, n = 3$ $a n d n = 4 . W h a t i s t h e s m a l l e s t p o s i t i v e$ $i n t e g e r w i t h 12 p r o p e r f a c t o r s ?$ $W h a t i s t h e s m a l l e s t p o s i t i v e i n t e g e r$ $w i t h a t l e a s t 12 p r o p e r f a c t o r s ?$ $(A p r o p e r f a c t o r o f a p o s i t i v e n u m b e r N$ $i s p o s i t i v e n t e g e r M s u c h t h a t M \neq 1 a n d M \neq N .)$

Question Number 1895 Answers: 2 Comments: 5

Let us generalise the result of taking the inverse tangent of a complex number to the form tan^(−1) (c+id)=a+ib where a,b,c,d∈R and i=(√(−1)). Determine a and b respectively in terms of c and d.

$L e t u s g e n e r a l i s e t h e r e s u l t o f t a k i n g t h e i n v e r s e t a n g e n t o f a c o m p l e x n u m b e r$ $t o t h e f o r m$ ${t a n}^{- 1} (c + i d) = a + i b$ $w h e r e a, b, c, d \in R a n d i = \sqrt{- 1} . D e t e r m i n e a a n d b r e s p e c t i v e l y i n t e r m s$ $o f c a n d d .$

Question Number 1776 Answers: 0 Comments: 0

P_k ={x∈N,n∈N:x>0,Σ_(n∣x) 1=k} C=∪_(k≥3) P_k {0,1}∪C∪P=N proof or give a counter example that (x,y,z)∈C^3 ,x^2 ∣yz⇒x∣y∨x∣z

$P_{k} = {x \in N, n \in N : x > 0, \sum_{n ∣ x} 1 = k}$ $C = \underset{k ⩾ 3}{\cup} P_{k}$ ${0, 1} \cup C \cup P = N$ $proof or give a counter example that$ $(x, y, z) \in C^{3}, x^{2} ∣ y z \Rightarrow x ∣ y \lor x ∣ z$

Question Number 1700 Answers: 1 Comments: 2

Show that (7!)^(1/7) <(8!)^(1/8) . Also show that (√(100001))−(√(100000))<(1/(2(√(100000)))) .

$S h o w t h a t {(7!)}^{\frac{1}{7}} < {(8!)}^{\frac{1}{8}} .$ $A l s o s h o w t h a t$ $\sqrt{100001} - \sqrt{100000} < \frac{1}{2 \sqrt{100000}} .$

Question Number 1164 Answers: 2 Comments: 1

How many five digit numbers exist such that the sum of their digits equals 43? How many exist if the sum is 39?

$H o w m a n y f i v e d i g i t n u m b e r s$ $e x i s t s u c h t h a t t h e s u m o f t h e i r$ $d i g i t s e q u a l s 43 ?$ $H o w m a n y e x i s t i f t h e s u m i s$ $39 ?$

Question Number 1156 Answers: 1 Comments: 1

Determine the general solution of the following linear diophantine equation for ∀N∈Z^+ ,m∈Z^+ : 8N=81m+65 .

$D e t e r m i n e t h e g e n e r a l s o l u t i o n$ $o f t h e f o l l o w i n g l i n e a r d i o p h a n t i n e$ $e q u a t i o n f o r \forall N \in Z^{+}, m \in Z^{+} :$ $8 N = 81 m + 65 .$

Question Number 795 Answers: 1 Comments: 0

what is the last digit of 7^((7^((7....)) )) the number of 7′s is 1001

$w h a t i s t h e l a s t d i g i t o f$ $7^{(7^{(7 . . . .)})}$ $t h e n u m b e r o f 7^{'} s i s 1001$

Question Number 785 Answers: 1 Comments: 0

Prove that if two numbers are chosen at random then the probability that their sum is divisible by n is (1/n).

$P r o v e t h a t i f t w o n u m b e r s a r e c h o s e n$ $a t r a n d o m t h e n t h e p r o b a b i l i t y t h a t$ $t h e i r s u m i s d i v i s i b l e b y n i s \frac{1}{n} .$

Question Number 772 Answers: 0 Comments: 4

prove that 5555^(2222) +2222^(5555) is divisible by 7

$p r o v e t h a t$ $5555^{2222} + 2222^{5555}$ $i s d i v i s i b l e b y 7$

Question Number 769 Answers: 1 Comments: 1

Prove that product of any n consecutive integers is divisiblr by n!

$P r o v e t h a t p r o d u c t o f a n y n c o n s e c u t i v e i n t e g e r s$ $i s d i v i s i b l r b y n!$

Question Number 765 Answers: 1 Comments: 0

List all primes p for which p+2 and p+4 are also primes.

$List all primes p for which p + 2 and p + 4$ $are also primes .$

Question Number 751 Answers: 1 Comments: 0

Determine the number of integral factors of 105840, excluding 1 and 105840.

$D e t e r m i n e t h e n u m b e r o f i n t e g r a l$ $f a c t o r s o f 105840, e x c l u d i n g 1$ $a n d 105840 .$

Question Number 591 Answers: 1 Comments: 0

Prove by induction on n, for n≥2, u_n ≥ 2^3^(n−1) for the sequence {u_n } defined by the recurrence relation u_1 =1 u_(n+1) =(u_n +(1/u_n ))^3 , n≥1 .

$P r o v e b y i n d u c t i o n o n n, f o r n ⩾ 2,$ $u_{n} ⩾ 2^{3^{n - 1}}$ $f o r t h e s e q u e n c e {u_{n}} d e f i n e d b y$ $t h e r e c u r r e n c e r e l a t i o n$ $u_{1} = 1$ $u_{n + 1} = {(u_{n} + \frac{1}{u_{n}})}^{3}, n ⩾ 1 .$

Question Number 572 Answers: 1 Comments: 0

xy=6(x+y) x^2 +y^2 =325 x=? y=?

$x y = 6 (x + y)$ $x^{2} + y^{2} = 325$ $x = ?$ $y = ?$

Question Number 558 Answers: 1 Comments: 0

Determine the complex number z such that z^2 =tanx+icosx, in the form z=p+ik, p,k∈R.

$D e t e r m i n e t h e c o m p l e x n u m b e r$ $z s u c h t h a t z^{2} = t a n x + i c o s x,$ $i n t h e f o r m z = p + i k, p, k \in R .$

Question Number 552 Answers: 1 Comments: 0

proof that log_2 3 is irrational

$p r o o f t h a t \log_{2} 3 i s i r r a t i o n a l$

Pg 20 Pg 21 Pg 22 Pg 23 Pg 24 Pg 25 Pg 26 Pg 27

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