Download Number Theory

Park Forest Math Team
Meet #3
Number
Theory
Self-study Packet
Problem Categories for this Meet:
1. Mystery: Problem solving
2. Geometry: Angle measures in plane figures including supplements and
complements
3. Number Theory: Divisibility rules, factors, primes, composites
4. Arithmetic: Order of operations; mean, median, mode; rounding; statistics
5. Algebra: Simplifying and evaluating expressions; solving equations with 1 unknown
including identities
Important information you need to know about NUMBER THEORY…
Bases, Scientific Notation
Number Bases
• We use the base-ten number system where each digit is a power of ten.
This means that every digit in the right column is worth 1, the 2nd (from the right)
worth 10, the 3rd worth 100, the 4th worth 1000, the 5th worth 10,000 and so on.
•
In any number base, n, the place values are always
and continue on as our system does.
•
To convert a number in another base into base 10, first find the value of each
place. Add all the values together. Example: Convert 231base 4 to base 10. The
far right’s place value is 40 or 1. The middle’s is 41 or 4. The left’s is 42 or 16.
There is 1 “1” (totaling 1), 3 “4’s” (totaling 12) and 2 “16’s” (totaling 32). If you
add the individual totals, 1 + 12 + 32, you get 45. 45 is the base 10 equivalent of
231base 4.
•
To convert a number from base 10 to another number system, first find out the
value of each digit in another system. Example: Convert 37base 10 to base 5.
Base 5’s place values are worth
, or 25, 5, and 1. Starting from the left,
there is one 25 in the base ten number 37 and there are 12 left over. There are
two 5s in the base ten number 12 with 2 left over. There are 2 ones in the base
ten number two. Therefore, 37base 10 is 122 in base 5 .
•
In any number base, the largest digit is one less than the number base itself. For
example, in base 10, the largest digit is 10-1 or 9. In base 6, the largest digit is 61, or 5.
•
You can multiply, divide, add, or subtract in another number base as long as you
remember to borrow or carry in that number base. You can also convert the
number to base 10, perform the numerical operation, and then convert back to
the original number base.
Scientific Notation
•
The form for Scientific Notation is
where a is a number between 1 and 10
(
) multiplied by 10 to an integral power.
•
When working with complex scientific notation problems, remember to crossreduce and make the problem a lot simpler!
Category 3
Number Theory
Meet #3 - January, 2014
50th anniversary edition
1) The numeral 1101001 is written in base 2. Write it in base 10.
2) The average strep bacterium has a diameter of about 90 x 10-6
of a meter while the average flu virus has a diameter of about
0.2 x 10-4 of a meter. How many times greater is the diameter of
an average strep bacterium than an average flu virus? Express your
answer in scientific notation.
3) Evaluate. Write your answer in scientific notation.
72 x 107 x 240 x 10-6 ÷ 0.0008 x 10-2
1.2 x 100 0.09 x 105
400 x 1012
Answers
1)
base 10
2)
3)
www.imlem.org
Solutions to Category 3
Number Theory
Meet #3 - January, 2014
Answers
1) from right to left:
(1x20 ) + (0x21 ) + (0x22 ) + (1x23 ) + (0x24 ) + (1x25 ) + (1x26 )
1)
105
= 1 + 0 + 0 + 8 + 0 + 32 + 64
= 105
2) 4.5 x 100
3) 8 x 1020
90 x 10-6
0.2 x 10-4
2) Divide the larger by the smaller:
=
9.0 x 10-5
2 x 10-5
= 4.5 x 100
72 x 107 240 x 10-6 0.0008 x 10-2
x
÷
5
0
0.09 x 10
1.2 x 10
400 x 1012
3)
=
7.2 x 108 2.4 x 10-4 8 x 10-6
x
÷
1.2 x 100
9 x 103
4 x 1014
=
7.2 x 108 2.4 x 10-4 4 x 1014
x
x
1.2 x 100
9 x 103
8 x 10-6
=
7.2 x 2.4 x 4 x 108 x 10-4 x 1014
1.2 x 9 x 8 x 100 x 103 x 10-6
= 0.8 x 1021
= 8 x 1020
www.imlem.org
Meet #3
January 2012
Category 3 – Number Theory
1. Solve the following Binary (Base ) problem. Give your answer in base
2. If stretched out, a DNA molecule can measure
meter is
nano-meters (a nano-
of a meter).
The diameter of Earth is
kilometers (a kilometer is
meters).
How many stretched-out DNA molecules can we fit in the diameter?
Express your answer in scientific notation.
3. All the numbers in this problem are in base . Your answer should also be
expressed in base .
Answers
1. _______________
2. _______________
3. _______________
www.imlem.org
.
Meet #3
January 2012
Answers
Solutions to Category 3 – Number Theory
1. We can do this in
steps:
1.
First:
2.
or
(simply adding two zeroes).
Then:
3.
This Binary number stands for
2. Converting both measures to meters, the answer will be:
Note that
is not a valid scientific notation (
but it’s
easier to see the division this way.
3. It is easiest to translate the numbers to base
, solve, and then translate the
answer back to base :
[Note the similarity to the following base
equality:
where we substituted the digits appropriately].
www.imlem.org
Category 3 - Number Theory
Meet #3, January 2010
1. Evaluate the following expression. Write your result in scientific notation.
2.3 ∙ 107 ∙ (4.8 ∙ 10−4 )
6 ∙ 105 + 9 ∙ 104
2. Find x if x satisfies the equation:
1,071𝐵𝑎𝑠𝑒
10
= 3,060𝐵𝑎𝑠𝑒
𝑥
3. Solve this Binary (base two) problem. Express your answer in Binary (base two).
11,111,111 − 101 ∙ 101,000 =?
Answers
1. _______________
2. _______________
3. _______________
www.imlem.org
Solutions to Category 3 - Number Theory
Meet #3, January 2010
1.
2.3∙10 7 ∙ 4.8∙10 −4
6∙10 5 +9∙10 4
=
2.3∙4.8∙10 7−4
6.9∙10 5
=
4.8∙10 3
3∙10 5
=
1.
2.
3.
Answers
1.6×10-2
7
110,111
1.6 ∙ 103−5 = 1.6 ∙ 10−2
2. The key is to realize that since in base x the number ends with a ‘0’, then x is a factor
of our number (1,071). That is true for any base (as an example, in base 10, numbers
ending with a ‘0’ are divisible by 10…). Factorization yields 1,071 = 3 × 7 × 51, so
obvious candidates are 3 and 7. [Other factors, like 51 or 3 × 7 = 21 would have been
possible candidates, but since (in any base) 3,060 > 1,071 it should be clear that
𝑥 < 10]. So trying our 2 candidates: 1,071𝐵𝑎𝑠𝑒 10 = 110,200𝐵𝑎𝑠𝑒 3 = 3,060𝐵𝑎𝑠𝑒 7
we conclude that 𝑥 = 7.
Another way to think about it is to write 3𝑥 3 + 6𝑥 =
1,071 and to try some (obviously odd!) values for x, as we know 𝑥 < 10.
3. We can do the Binary arithmetic.
First:
101
× 101,000
101,000
000,000
101,000
11,001,000
And then:
-
11,111,111
11,001,000
110,111
Or we can translate everything to
decimal:
𝟏𝟎𝟏 = 22 + 20 = 4 + 1 = 5
𝟏𝟎𝟏, 𝟎𝟎𝟎 = 25 + 23
= 32 + 8
= 40
𝟏𝟏, 𝟏𝟏𝟏, 𝟏𝟏𝟏
= 𝟏𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎 − 𝟏
= 28 − 1 = 256 − 1
= 255
So our problem becomes 255 − 5 ∙ 40
And the answer is
55 = 32 + 16 + 4 + 2 + 1
= 𝟏𝟏𝟎, 𝟏𝟏𝟏
www.imlem.org
Category 3
Number Theory
Meet #3, January 2008
1.
Express the base 4 number 32134 as a base ten number.
3213base four = _______base ten
2.
A jar of sand has 910g of sand inside of it. Each grain of sand weighs
.013g. How many grains of sand are in the jar? Express your answer in
scientific notation.
3.
What is the base seven value of this subtraction problem?
235417 – 56357 = _______base 7
Answers
1. __________base 10
2. _______________
3. __________base 7
Solutions to Category 3
Number Theory
Meet #3, January 2008
Answers
1. 32134 = 3(43)+2(42)+1(41)+3(40) = 3(64)+2(16)+1(4)+3(1) =
192 + 32 + 4 + 3 = 231 base 10
1. 231
2. 7 10
or 7.0 10
3. 146037
2.
.
70000 . 3. To subtract 56357 from 235417 you do the exact same thing
you would do if they were both base ten numbers except that
when you need to borrow, you do not borrow 10, you borrow a 7.
So in the first step of the problem when you need to subtract the
5 from the 1 you would normally borrow from the 4 and add 10
to the 1. Now you will borrow from the 4 and add 7 to the 1.
The units digit becomes 8 – 5 = 3. Continuing this subtraction
you get 146037.
Category 3
Number Theory
Meet #3, January 2006
1. Express the base three number 121212 as a base nine number.
121212 base three = ______ base nine
2. Evaluate the following expression. Write your result in scientific notation.
(1.32 × 10 )(1.4 × 10 )
(1.1 × 10 )
−7
6
−9
3. Solve the following base-four equation for x. Write your result in base four.
Remember that all numbers shown in the equation are base-four numbers.
2(13x + 22) −12 = 3200
Answers
1. __________ base nine
2. _______________
3. __________ base four
www.imlem.org
Solutions to Category 3
Number Theory
Meet #3, January 2006
Answers
1. 555base niine
2. 1.68 ×10 8
3. 33base four
1. The first six place values in base three are 243, 81, 27, 9, 3,
and 1. This means the six-digit number 121212 in base three is
equal to 1 × 243 + 2 × 81+ 1 × 27 + 2 × 9 + 1 × 3 + 2 ×1, which is
455 in base ten. We can convert the expression above directly
to base nine, however, without using the base-ten value. The
first three place values of base nine are 81, 9, and 1. We can
think of the 243 as 3 more 81’s, the 27 as 3 more 9’s, and the 3
as 3 more 1’s. In all, we have five 81’s, five 9’s, and five 1’s,
so the base nine number must be 555base nine. If you do convert
455base ten to base nine, you would subtract five 81’s, which
leaves 50. Then you would subtract five 9’s, which leaves 5.
Again, you get 555base nine.
2. The expression can be solved as follows:
(1.32 ×10 )(1.4 ×10 ) = 1.32 ×1.4 × 10 ×10
1.1
10
(1.1 ×10 )
−7
6
6
−7
−9
−9
1.848
×10 6−7−(−9)
1.1
= 1.68 ×10 8
=
3. The equation is solved directly in base four on the left and converted to base
ten and back on the right.
2(13x + 22) −12 = 3200
32x + 110 −12 = 3200
32x + 32 = 3200
32(x + 1) = 3200
x + 1 = 100
x = 100 −1
x = 33 base four
2(13x + 22) −12 = 3200 base four
2(7x + 10) − 6 = 224 base ten
14 x + 20 − 6 = 224
14 x + 14 = 224
14 (x + 1) = 224
x + 1 = 16
x = 15 base ten
x = 33 base four
www.imlem.org
Category 3
Number Theory
Meet #3, January 2004
1. Solve the following base five equation for n. Express your result in base five.
2343base five + nbase five = 4032 base five
2. A box of twelve eggs is commonly refered to as a “dozen” eggs. Less common
are the words “gross”, which means a dozen dozen, and “great gross”, which
means a dozen dozen dozen. How many eggs are there in two great gross five
gross nine dozen three?
3. Evaluate the following expression. Write your result in scientific notation.
(3.8 ×10 )(1.2 ×10 )
(5.7 ×10 )
−4
17
−13
Answers
1. _______________
2. _______________
3. _______________
www.Imlem.org
Solutions to Category 3
Number Theory
Meet #3, January 2004
Answers
1. 1134
2. 4287
3. 8 ×10 25
1. To find the base five value of n that satisfies the equation
2343base five + nbase five = 4032 base five , we need to subtract 2343
from 4032 in base five. When we “borrow”, we need to
remember that we get five instead of ten.
The computation shown at right does not show
evidence of the borrowing that most people do to
4032
get this result. For example, that first 2 minus 3
in the ones place would become 7 minus 3 when −2343
we borrow a set of five from the column to the
1134
left. The 3 in the second column would become a
2 and then we would have to borrow again.
We should confirm that n is 1134 by returning to
the original equation and checking the addition.
2343base five + 1134 base five = 4032 base five
2. Before we can interpret the phrase “two great gross five gross nine dozen
three”, we need to establish the base ten values of these base twelve words. Dozen
means 12, gross means 12 × 12 = 144, and great gross means 12 × 12 × 12 = 144 ×
12 = 1728. Now we can translate the word expression to the numerical expression
2 ×1728 + 5 ×144 + 9 ×12 + 3 and evaluate it. We get
2 ×1728 + 5 ×144 + 9 ×12 + 3 = 3456 + 720 + 108 + 3 = 4287 eggs.
3. Since the expression consists of multiplication and division, we can rearrange
3.8 ×10 17 )(1.2 ×10−4 ) 3.8 ×1.2 1017 ×10 −4
(
them as follows:
=
×
. The number part
5.7
10 −13
(5.7 ×10−13 )
2 ×1.9 ×1.2 2 ×1.2
can be reduced to
=
= 0.8 and the power of ten becomes
3 ×1.9
3
1017 ×10 −4
= 10 17−4−(−13) = 10 26 . We now have 0.8 ×10 26 = 8 × 10 25 .
−13
10
www.Imlem.org