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Simple Facts about Numbers
Answer each question:
1. It has the fewest factors of any number.
2. It is the smallest number that has only two factors.
3. This one-digit number has the same number of factors as the number
6.
4. It is the smallest number that has six factors.
5. It is the smallest three-digit perfect square number.
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Simple Facts about Numbers
6. It is the smallest three-digit number that has only two factors.
7. It is the largest two-digit perfect square number that has three factors.
8. It is the smallest three-digit perfect square that has nine factors.
9. The factors of this number are 1, 2, 3, 4, 6, 12.
10. It is the smallest perfect square that has nine factors.
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Who am I?
Answer each question:
1. I am the largest one-digit prime number.
2. I am the smallest two-digit prime number.
3. I have only one factor.
4. I am the only even prime number.
5. I am the largest two-digit prime number.
6. The number 59 and I are the only two prime numbers between 50 and
60.
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Who am I?
7. If I am the units digit of any number, then the number is divisible by
10.
8. I am not the number 1, but I am a factor of 60 and 35.
9. I am a one-digit number. If you add all of my factors, the sum equal
12.
10. I am the largest one-digit number that has only three factors.
11. I am the first prime number after 100.
12. I am the largest composite number before 100.
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Cubes and Squares
Find the number related to squares and cubes:
1. Find the only one-digit number that is both a square and a cubic
number.
2. Find the only two-digit number that is both a square and a cubic
number.
3. Find the only three-digit number that is both a square and a cubic
number.
4. Find a two-digit number that one more than a square number and
one less than a cubic number.
5. Find the smallest two-digit square number that can be written as a
sum of two square numbers.
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Cubes and Squares
Find the number related to squares and cubes:
6. Find the smallest three-digit square number that is the sum of two
square numbers.
7. Find the only one-digit number that can be written as the sum of two
different cubic numbers.
8. Find the smallest two-digit number that can be written as the sum of
two cubic numbers.
9. Find the largest two-digit number that can be written as the sum of
two different cubic numbers.
10. Find a two-digit number that one less than a square number, and
when doubled is one less than a square number.
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Filling Numbers
Complete the problems by substituting x, y , z, w with the numbers
2, 4, 6, 8 to find the answer described on the right. Using only proper
fractions in the problems and answers. Each number must be used once in
each problem:
1.
x
y
+
z
w
=
largest sum
2.
x
y
+
z
w
=
smallest sum
3.
x
y
−
z
w
=
largest difference
4.
x
y
−
z
w
=
smallest difference
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Filling Numbers
Complete the problems by substituting x, y , z, w with the numbers
2, 4, 6, 8 to find the answer described on the right. Using only proper
fractions in the problems and answers. Each number must be used once in
each problem:
5.
x
y
×
z
w
=
largest product
6.
x
y
×
z
w
=
smallest product
7.
x
y
÷
z
w
=
largest quotient
8.
x
y
÷
z
w
=
smallest quotient
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Think about This
Find the fractions described below. For some problems, there may be more
than one answers, but you need only give one. (Remember that whole
numbers may also be written as fractions with a denominator of 1.):
1. Find two fractions. There product is less than their sum.
2. Find two fractions. Their quotient is larger than their difference.
3. Find a fraction that is equal to its reciprocal.
4. Find two different fractions with a numerator of 1 that have the
largest sum.
5. Find a fraction that can be expressed as a repeating decimal, but
whose reciprocal terminates.
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Think about This
Find the fractions described below. For some problems, there may be more
than one answers, but you need only give one. (Remember that whole
numbers may also be written as fractions with a denominator of 1.):
6. Find a simplified fraction and its reciprocal that both can be
expressed as terminating decimals.
7. Find two proper fractions whose sum, difference, product, and
quotient are between 0 and 1.
8. Find two improper fractions whose sum and product are greater than
1, and whose difference and quotient are less than 1.
9. Find three fractions whose sum is less than 1.
10. Find three simplified fractions whose sum is 1.
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Human Body Statistics
Write each italicized number fact about the human body in scientific
notation.:
1. According to one theory, most human beings live for about 2.4 billion
heartbeats.
2. The human body contains 7.5 trillion cells. Each day 2 billion cells die
and are replaced.
3. There are about 1.5 gallons of blood in men and 0.875 gallons in
women.
4. The adult human body has about 60000 miles of blood vessels.
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Human Body Statistics
Write each italicized number fact about the human body in scientific
notation.:
5. If the air sacs, or alveoli, in the lungs were flattened out, that would
cover between 600 and 1000 square feet.
6. The human body responds to warmth and cold within 0.1 to 0.2
seconds.
7. The reaction times for smell and pain are about 0.3 and 0.7 seconds.
8. Impulses can travel through the human nervous system as fast as 223
miles per hour.
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Celestial Facts
Write each number in standard form.:
1. Mean distance from Mercury to the sun: 3.6 × 107 miles =
2. Mean distance from Earth to the sun: 9.296 × 107 miles =
3. Mean distance from the Moon to the Earth: 2.4 × 105 miles =
4. Mean distance from Pluto to the sun: 3.666 × 109 miles =
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Celestial Facts
Write each number in standard form.:
5. Sun’s temperature at its core: 2.7 × 107 ◦ F =
6. Sun’s temperature at its surface: 8.7 × 103 ◦ F =
7. Radius of the sun: about 4.3 × 105 miles =
8. Speed of light (in a vacuum): about 1.86 × 105 miles per second =
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Swimming Pool
I have an octagonal swimming pool. Its eight sides are, consecutively, 10
meters, 20 meters, 30 meters, 40 meters, 50 meters, 60 meters, 70 meters,
and 80 meters long. All the pool’s angles are right angles. What is the
surface area of the pool in square meters?
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Some One
If I add
1 + 11 + 111 + 1111 + · · · + 111 . . . 111,
where the last number consists of digit 1 repeated 96 times, how many 1’s
will be in the result?
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How many 1’s?
If I type a set of numbers from
1, 2, 3, . . . , 1000000,
find the number of times that the digit 1’s appeared in the sequence.
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Several 1’s
1. Let n be a positive integer and put Rn = 19 · (10n − 1). Thus Rn is an
integer. Determine all integers n for which Rn is a perfect square
integer.
2. Prove that there are infinitely many integers n such that 7 is a factor
of Rn .
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The Five Numbers
The teacher wrote five real numbers on the back of the blackboard. Then
on the front of the board he wrote the numbers 6, 7, 8, 8, 9, 9, 10, 10, 11,
and 12. These numbers are the ten possible sums of two of the five hidden
numbers. What are the five hidden numbers? List the numbers in
ascending order.
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Fractions
1. Evaluate:
1 + 20114 + 20124
1 + 20112 + 20122
2. Find a simple formula for
1 + x 4 + (x + 1)4
1 + x 2 + (x + 1)2
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Integer solutions
1. Find all ordered pairs of positive integers (x, y ) for which
1
1
1
+ =
x
y
13
2. Let p be a prime. Find all ordered pairs of positive integers (x, y ) for
which
1
1
1
+ =
x
y
p
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Pigeonhole Principle
1. Prove that among any 51 numbers chosen randomly from
{1, 2, 3, . . . , 100} there are two numbers which are relatively prime.
2. Prove that among any 51 numbers chosen randomly from
{1, 2, 3, . . . , 100} there are two numbers x, y , such that x is a factor
of y or y is a factor of x.
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Intermediate Value Theorem
1. Prove that there exist 1000 consecutive positive integers, such that
there are exactly 10 primes among them.
2. The following is TRUE or FALSE: There exist 101000 consecutive
positive integers, such that there are exactly 10 perfect square
numbers among them.
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Pairing or Not Pairing
1. Show that it is possible to pair off the numbers 1, 2, 3, . . . , 10 so that
the sum of each of the five pairs are five different prime numbers.
2. Is it possible to pair off the numbers 1, 2, 3, . . . , 20 so that the sum of
each of the ten pairs are ten different prime numbers?
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Game 24
1. Find the number of sets a, b, c, d of integers with
0 ≤ a ≤ b ≤ c ≤ d ≤ 9.
2. Find the number of sets a, b, c, d of integers with
0 ≤ a ≤ b ≤ c ≤ d ≤ 9 satisfying the Game 24.
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Game 24
3. Check the following sets satisfying the Game 24 1, 2, 3, 4; 2, 3, 4, 5;
a, a + 1, a + 2, a + 3 (a = 1, 2, . . . , 6);
1, 2, 7, 7; 2, 3, 8, 8; 2, 6, 8, 9.
4. Find all x such that x, x, x, x satisfies the Game 24.
5. Find all pairs of integers x, y with x < y and x, x, y , y satisfies the
Game 24.
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FiFa World Cup 2010
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Tournaments
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Euclid and the GCF
More than two thousand years ago, Euclid, the Greek mathematician,
devised a method to find the GCF (greatest common factor) of the two
numbers. You can use this method today. Just follow these steps:
1. Divide the larger number by the smaller number.
2. Divide the smaller number by the remainder in the first step.
3. Repeat this process until there is no remainder.
4. The last divisor is the GCF of the original number.
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Euclid and the GCF
Here is an example. Find the GCF of 224 and 78. Follow Euclid’s steps:
1. Divide 224 by 78. The answer is 2 R68.
2. Divide 78 by 68. The answer is 1 R10.
3. Divide 68 by 10. The answer is 6 R8.
4. Divide 10 by 8. The answer is 1 R2.
5. Divide 8 by 2. The answer is 2; 2 is the GCF of 224 and 78.
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Euclid and the GCF
Use Euclid’s method to find the GCF of each pair of numbers.
1. 105, 27
2. 40, 27
3. 82, 96
4. 1777, 1855
5. 645541, 1512227
6. 6619237, 11111111
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Graph Theory
1. Let a, b, c, d be positive integers, a < b < c < d,
GCF (a, b, c, d) = 1, and M = a + b + c + d. Let G (a, b, c, d) be a
graph with V = {a, b, c, d} as its vertex set and x, y ∈ V , x 6= y , and
xy is an edge of G (a, b, c, d) if and only if x + y | M.
1.1 Prove that there are infinitely many sets {a, b, c, d} such that
G (a, b, c, d)
contains no edges.
1.2 Prove that there are infinitely many sets {a, b, c, d} such that
G (a, b, c, d)
contains 3 edges.
1.3 Find all sets {a, b, c, d} such that G (a, b, c, d)
contains 4 edges.
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Graph Theory
2 Let a, b, c, d, e be positive integers with a < b < c < d < e,
GCF (a, b, c, d, e) = 1, and M = a + b + c + d + e. The graph
G (a, b, c, d, e) can be defined similarly as above.
2.1 Prove that there are infinitely many sets {a, b, c, d, e} such that
G (a, b, c, d, e) contains 6 edges.
2.2 What is the maximum number of edges that G (a, b, c, d, e) can have?
(Proof your answer)
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