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Prime number
A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. This theoremrequires excluding 1 as a prime. -------------------------------------------------

Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of theintegers. Number theorists study prime numbers (which, when multiplied, give all the integers) as well as the properties of objects made out of integers (such as rational numbers) or defined as generalizations of the integers (such as, for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (diophantine geometry). Questions in number theory are often best understood through the study ofanalytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter -------------------------------------------------

Prime number theorem
From Wikipedia, the free encyclopedia
"PNT" redirects here. For other uses, see PNT (disambiguation). In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers. Informally speaking, the prime number theorem states that if a random integer is...

...10th Real Numbers test paper
2011
1.
Express 140 as a product of its prime factors
2.
Find the LCM and HCF of 12, 15 and 21 by the prime factorization method.
3.
Find the LCM and HCF of 6 and 20 by the prime factorization method.
4.
State whether13/3125 will have a terminating decimal expansion or a non-terminating repeating
decimal.
5.
State whether 17/8 will have a terminating decimal expansion or a non-terminating repeating
decimal.
6.
Find the LCM and HCF of 26 and 91 and verify that LCM × HCF = product of the two numbers.
7.
Use Euclid’s division algorithm to find the HCF of 135 and 225
8.
Use Euclid’s division lemma to show that the square of any positive integer is either of the form
3m or 3m + 1 for some integer m
9.
Prove that √3 is irrational.
10. Show that 5 – √3 is irrational
11. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some
integer.
12. An army contingent of 616 members is to march behind an army band of 32 members in a parade.
The two groups are to march in the same number of columns. What is the maximum number of
columns in which they can march?
13. Express 156 as a product of its prime factors.
14. Find the LCM and HCF of 17, 23 and 29 by the prime factorization method.
15. Find the HCF and LCM of 12, 36 and 160,...

...Section 4.1
Divisibility and Modular Arithmetic
87
CHAPTER 4
Number Theory and Cryptography
SECTION 4.1
Divisibility and Modular Arithmetic
2. a) 1 | a since a = 1 · a.
b) a | 0 since 0 = a · 0.
4. Suppose a | b , so that b = at for some t , and b | c, so that c = bs for some s. Then substituting the ﬁrst
equation into the second, we obtain c = (at)s = a(ts). This means that a | c, as desired.
6. Under the hypotheses, we have c = as and d = bt for some s and t . Multiplying we obtain cd = ab(st),
which means that ab | cd , as desired.
8. The simplest counterexample is provided by a = 4 and b = c = 2.
10. In each case we can carry out the arithmetic on a calculator.
a) Since 8 · 5 = 40 and 44 − 40 = 4 , we have quotient 44 div 8 = 5 and remainder 44 mod 8 = 4.
b) Since 21 · 37 = 777, we have quotient 777 div 21 = 37 and remainder 777 mod 21 = 0 .
c) As above, we can compute 123 div 19 = 6 and 123 mod 19 = 9. However, since the dividend is negative
and the remainder is nonzero, the quotient is −(6 + 1) = −7 and the remainder is 19 − 9 = 10. To check that
−123 div 19 = −7 and −123 mod 19 = 10 , we note that −123 = (−7)(19) + 10.
d) Since 1 div 23 = 0 and 1 mod 23 = 1 , we have −1 div 23 = −1 and −1 mod 23 = 22 .
e) Since 2002 div 87 = 23 and 2002 mod 87 = 1, we have −2002 div 87 = −24 and 2002 mod 87 = 86 .
f) Clearly 0 div 17 = 0 and 0 mod 17 = 0.
g) We have 1234567 div 1001 = 1233 and 1234567 mod 1001 = 334.
h) Since...

...Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the expression for the given values of x and y. |x| |y| 1) + ; x = 2 and y = -4 x y A) 2 Simplify the algebraic expression. 2) -4(2x - 5) - 4x + 9 A) -12x + 29 Simplify the exponential expression. 3) (x3)6 A) 6x18 Rationalize the denominator. 3 4) 17 + 2 A) 51 - 2 3 13 B) 51 + 2 3 13 C) 3 51 + 17 34 3 D) 51 - 2 3 19 B) x9 C) x18 D) 6x3 B) 4x + 29 C) -12x - 11 D) 12x + 29 B) -1 C) 0 D) 1
Find the product. 5) (x - 3)(x2 + 3x + 9) A) x3 - 6x2 - 6x - 27 B) x3 + 27 C) x3 - 27 D) x3 + 6x2 + 6x - 27
Factor the trinomial, or state that the trinomial is prime. 6) 20x2 + 23x + 6 A) (20x + 3)(x + 2) B) (4x - 3)(5x - 2) C) (4x + 3)(5x + 2) D) Prime
Factor completely, or state that the polynomial is prime. 7) 28x2 y - 28y - 28x2 + 28 A) (2y - 7)(7x - 2)(7x + 2) C) (7y - 7)(2x - 2)(2x + 2) Solve the system by the addition method. 8) 3x + 7y = 40 3x + 2y = 50 A) {(-2, 18)} Solve and check the linear equation. 9) 2x - 4 + 5(x + 1) = -2x - 3 A) {- 2} B) {4 } 3 C) {4 } 9 D) {- 6} B) {(-18, 3)} C) {(-18, 7)} D) {(18, -2)} B) (28y - 28)x2 + 4(-7y + 7) D) (28x2 - 28)y + 7(4 - 4x2 )
1
Solve the equation. x x 10) 27 - = 2 7 A) { 243 } 14 B) {42} C) {3} D) { 243 } 2
First, write the value(s) that make the denominator(s) zero. Then solve the equation. x-1 x+9 11) +3 = 4x x A) x ≠ 0;...

...COMPARATIVE ANALYSIS OF DATA 1 (RULES) AND DATA 2 (ITEMS)
John Paul Llenos (Organizer)
Patricia Lorica (Secretary)
CED 02 – 601P
Language and Literature Assessment
Rules
Items
Error
Correct
Error
Correct
11 – 47
29 – 45
25 – 44
1 – 37
9 – 36
23 – 35
5 – 30
17 – 30
27 – 30
3 – 29
15 – 28
21 – 27
13 – 27
7 – 19
19 – 17
22 – 46
8 – 35
12 – 35
26 – 31
20 – 30
24 – 29
30 – 26
16 – 25
4 – 21
18 – 20
2 – 17
10 – 17
14 – 12
28 – 10
6 – 1
This paper aims to compare the data’s 1 and 2 through comparative analysis. It can be seen above the top items that obtained the most number of mistakes.
In rules data, there were 47 and 45 students who got items number 11 and 29 wrong. In the same manner with the items data, wherein 46 students got it wrong in item number 22. Items 11 and 29 hold Dangling Modifier as the correct answer. Therefore, error data articulates that Dangling Modifier is one of the grammar and style that needs special attention to the students subjected in this analysis.
In conclusion, though there is only a little difference between the two items. There’s much difference in the overall items between RULES data and ITEMS data. In rules data, most of the students got errors in most of the items compared to the items dat. We noticed that the items in rules data seem more difficult than the items in items data.
70. Dora Williams
WHEN Reuben Pantier ran away and...

...GRADE 5)
CHAPTER 1 (LARGE NUMBERS) ONE MARK QUESTIONS
1. 7000 lakh = _______________________ crore. a) 7 b) 70 c) 700 d) 7000
TWO MARK QUESTIONS
1. Write 700083460 in numerals and their number names in both the systems of numeration. 2. Write the smallest and the greatest numbers using each of the digits 4, 8, 0, 1, 7, 6, 5 only once.
CHAPTER 2 (ROUNDING NUMBERS AND ESTIMATION) ONE MARK QUESTIONS
1. The municipal corporation spent Rs. 25, 37, 981 on repairing the roads (round it to the nearest ten thousand). a) 25,40,000 b) 25,30,000 c) 26,00,000 d) 25,37,000 2. Which of the following numbers could be rounded to 9700? a) 9585 b) 9755 c) 9655 d) 9645
TWO MARK QUESTIONS
1. Round off the greatest 9 – digit number to the nearest ten – lakh. 2. Estimate: 63809 – 5523.
CHAPTER 3 (OPERATIONS ON LARGE NUMBERS) ONE MARK QUESTIONS
1. If 3900 kg of onions are put into sacks and each sack holds 30 kg, how many sacks are required? a) 30 b) 130 c) 117000 d) 3900
TWO MARK QUESTIONS
1. Find the product of the successor of the greatest 3 digit number and 999.
THREE MARK QUESTIONS
1. Raj won 35 tournaments. The prize money totalled up to Rs. 6, 47,500. If he received the same amount for every tournament, how much had he earned per tournament?
CHAPTER 4 (FACTORS AND MULTIPLES) ONE MARK QUESTIONS
1. The prime factorization of 27 is...