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Number TheoryQuestion and Answers: Page 26

Question Number 548 Answers: 1 Comments: 0

how many digits are in periodic part of (1/(60^(30) ))

$h o w m a n y d i g i t s a r e i n p e r i o d i c p a r t$ $o f \frac{1}{60^{30}}$

Question Number 518 Answers: 1 Comments: 0

Give the result of the following computation as an integer in the usual decimal form. ((303,000,000,000,303×3,300,000,033)/(1,000,100,010,001))

$G i v e t h e r e s u l t o f t h e f o l l o w i n g c o m p u t a t i o n a s a n i n t e g e r i n t h e u s u a l d e c i m a l$ $f o r m .$ $\frac{303, 000, 000, 000, 303 \times 3, 300, 000, 033}{1, 000, 100, 010, 001}$

Question Number 515 Answers: 1 Comments: 0

Find the smallest number greater than zero which can be written with ones and zeroes and is evenly divisble by 225.

$F i n d t h e s m a l l e s t n u m b e r g r e a t e r$ $t h a n z e r o w h i c h c a n b e w r i t t e n$ $w i t h o n e s a n d z e r o e s a n d i s e v e n l y d i v i s b l e$ $b y 225 .$

Question Number 514 Answers: 1 Comments: 0

Determine the smallest value of the form f(u,v)=((5v^2 +5u^2 +1)/(2u+v)) where u,v∈R^+ .

$D e t e r m i n e t h e s m a l l e s t v a l u e o f$ $t h e f o r m$ $f (u, v) = \frac{5 v^{2} + 5 u^{2} + 1}{2 u + v}$ $w h e r e u, v \in R^{+} .$

Question Number 513 Answers: 1 Comments: 0

What is the greatest common divisor of the 2010 digit and 2005 digit numbers below? 222...222 (2010 of twos) 777...777 (2005 of sevens)

$W h a t i s t h e g r e a t e s t c o m m o n$ $d i v i s o r o f t h e 2010 d i g i t a n d 2005 d i g i t$ $n u m b e r s b e l o w ?$ $222 . . . 222 (2010 o f t w o s)$ $777 . . . 777 (2005 o f s e v e n s)$

Question Number 503 Answers: 0 Comments: 1

proof or given a counter−example: if n∈N,n>1, exist a number k∈N k∈(0,n] such that n+k is prime.

$p r o o f o r g i v e n a c o u n t e r - e x a m p l e :$ $i f n \in N, n > 1, e x i s t a n u m b e r k \in N$ $k \in (0, n] s u c h t h a t n + k i s p r i m e .$

Question Number 475 Answers: 1 Comments: 0

proof or given a counter example: if p is prime and n∈N,1<n≤p then pn−1 is prime

$p r o o f o r g i v e n a c o u n t e r e x a m p l e :$ $i f p i s p r i m e a n d n \in N, 1 < n ⩽ p$ $t h e n p n - 1 i s p r i m e$

Question Number 471 Answers: 1 Comments: 0

proof or given a counter example: if n^2 is prime, then n∉Z

$p r o o f o r g i v e n a c o u n t e r e x a m p l e :$ $i f n^{2} i s p r i m e, t h e n n \notin Z$

Question Number 469 Answers: 1 Comments: 0

proof or given a counter−example if p is prime and a∈N then p∣(a+p)^p −a^p

$p r o o f o r g i v e n a c o u n t e r - e x a m p l e$ $i f p i s p r i m e a n d a \in N t h e n$ $p ∣ {(a + p)}^{p} - a^{p}$

Question Number 443 Answers: 0 Comments: 1

Prove or disprove that minimum value of n which satisfies the equation 10^n ≡1(mod 7^p ) is n=6×7^(p−1) .

$Prove or disprove that minimum value$ $of n which satisfies the equation$ $10^{n} \equiv 1 (\mod 7^{p}) is n = 6 \times 7^{p - 1} .$

Question Number 427 Answers: 1 Comments: 1

How many digits are present in periodic part for decimal expansion of (1/7^(11) )?

$How many digits are present in$ $periodic part for decimal expansion$ $of \frac{1}{7^{11}} ?$

Question Number 425 Answers: 1 Comments: 0

x^2 +y^3 =z^4 have integer solutions?

$x^{2} + y^{3} = z^{4} have integer solutions ?$

Question Number 403 Answers: 1 Comments: 0

(1/3^(2005) ) have how many digits in periodic part?

$\frac{1}{3^{2005}} have how many digits in periodic part ?$

Question Number 400 Answers: 1 Comments: 0

(1/3^(20) ) have how many digits in periodic part?

$\frac{1}{3^{20}} have how many digits in periodic part ?$

Question Number 348 Answers: 0 Comments: 1

a_n =1−(2/(3−(4/(5−n)))),n≡1(mod 2) a_n =1−(2/(3−(4/(5−(6/n))))),n≡0(mod 2) lim_(n→∞) a_n =^? 1

$a_{n} = 1 - \frac{2}{3 - \frac{4}{5 - n}}, n \equiv 1 (\mod 2)$ $a_{n} = 1 - \frac{2}{3 - \frac{4}{5 - \frac{6}{n}}}, n \equiv 0 (\mod 2)$ $\lim_{n \to \infty} a_{n} \overset{?}{=} 1$

Question Number 347 Answers: 0 Comments: 0

1−(2/(3−(4/(5−⋱))))=^? 1

$1 - \frac{2}{3 - \frac{4}{5 - ⋱}} \overset{?}{=} 1$

Question Number 215 Answers: 1 Comments: 0

evaluate 1+2^3 +3^4^5 (mod 10)

$evaluate$ $1 + 2^{3} + 3^{4^{5}} (\mod 10)$

Question Number 190 Answers: 1 Comments: 0

solve 30x≡50(mod 40)

$solve 30 x \equiv 50 (\mod 40)$

Question Number 187 Answers: 1 Comments: 0

solve for integer x,y 4x^2 −9y^2 =6

$solve for integer x, y$ $4 x^{2} - 9 y^{2} = 6$

Question Number 185 Answers: 1 Comments: 0

solve 10x≡25(mod 15)

$solve 10 x \equiv 25 (\mod 15)$

Question Number 177 Answers: 1 Comments: 0

solve 12x≡34(mod 56)

$solve 12 x \equiv 34 (\mod 56)$

Question Number 176 Answers: 1 Comments: 0

find the integer solution of 3x+4y=5

$find the integer solution of 3 x + 4 y = 5$

Question Number 170 Answers: 1 Comments: 0

solve 12x≡25(mod 69)

$solve 12 x \equiv 25 (\mod 69)$

Question Number 139 Answers: 1 Comments: 0

express the folowing in the0.6(bar)= form of p/q ;where p and q are integers and q is not =0 0.6^

$e x p r e s s t h e f o l o w i n g i n t h e 0.6 (b a r) =$ $f o r m o f p / q; w h e r e p a n d q$ $a r e i n t e g e r s a n d q i s n o t = 0$ $0 . \bar{6}$

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