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. : 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 2 nd (from the right) worth 10, the 3 rd worth 100, the 4 th worth 1000, the 5 th 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 231 base 4 to base 10. The far right s place value is 4 0 or 1. The middle s is 4 1 or 4. The left s is 4 2 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 231 base 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 37 base 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, 37 base 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 6-1, 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 Meet #3 - January, 2016 1) 10110 is a base three numeral. What is its value in base 10? 2) Express the answer to the calculation below in scientific notation: 3) The decimal (base 10) numeral 427.608 means 2 1 0-1 -2-3 (4x10 )+(2x10 )+(7x10 )+(6x10 )+(0x10 )+(8x10 ). The base 4 numeral 213.23 means. For each numeral, the "point" separates the whole number part from the fractional part. Express the base 8 numeral 53.21 as a base 2 numeral, including the point. 1) 2) 3)
Solutions to Category 3 Meet #3 - January, 2016 1) 10110 (base 3) = 81 + 9 + 3 = 93. 2) 1) 93 2) 3) 101011.010001 3) First expand and evaluate the base 8 numeral 53.21 from left to right): = 5(8) + 3(1) + 2(1/8) + 1(1/64) = 40 + 3 + 1/4 + 1/64 = 43 and 17/64 Since the denominator, 64, is a power of 2, we must express 17/64 as the sum of fractions whose denominators are powers of 2 and whose numerators are each less than 2. The fraction 1/4 uses up 16/64, leaving just 1/64 remaining. The whole number part, 43, can be expressed as the sum of these powers of 2: 32 + 8 + 2 + 1. The answer can be computed as = 101011.010001
Category 3 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 0.2 x 10-4 while the average flu virus has a diameter of about 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 107 240 x 10-6 0.0008 x 10-2 x 1.2 x 100 0.09 x 105 400 x 1012 1) base 10 2) 3)
Solutions to Category 3 Meet #3 - January, 2014 1) from right to left: (1x2 0 ) + (0x2 1 ) + (0x2 2 ) + (1x2 3 ) + (0x2 4 ) + (1x2 5 ) + (1x2 6 ) 1) 105 = 1+ 0 + 0 + 8 + 0 + 32 + 64 = 105 2) 4.5 x 100 3) 20 8 x 10 2) Divide the larger by the smaller: 90 x 10 0.2 x 10-6 -4 = 9.0 x 10-5 2 x 10-5 = 0 4.5 x 10 3) 72 x 107 240 x 10-6 0.0008 x 10-2 x 1.2 x 100 0.09 x 105 400 x 1012 = = = 7.2 x 10 8 2.4 x 10-4 8 x 10-6 x 1.2 x 10 0 9 x 10 3 4 x 10 14 7.2 x 10 8 2.4 x 10-4 4 x 10 14 x x 1.2 x 10 0 9 x 10 3 8 x 10-6 8-4 14 7.2 x 2.4 x 4 x 10 x 10 x 10 0 3-6 1.2 x 9 x 8 x 10 x 10 x 10 = 21 0.8 x 10 = 8 x 20 10
Category 3 Meet #3 January 2012 1. Solve the following Binary (Base ) problem. Give your answer in base. 2. If stretched out, a DNA molecule can measure nano-meters (a nanometer is 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. 1. 2. 3.
Meet #3 January 2012 Solutions to Category 3 1. We can do this in steps: First: (simply adding two zeroes). Then: 1. 2. or 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].
Category 3 - Meet #3, January 2010 1. Evaluate the following expression. Write your result in scientific notation. 2.3 10 7 (4.8 10 4 ) 6 10 5 + 9 10 4 2. Find x if x satisfies the equation: 1,071 Base 10 = 3,060 Base x 3. Solve this Binary (base two) problem. Express your answer in Binary (base two). 11,111,111 101 101,000 =? 1. 2. 3.
Solutions to Category 3 - Meet #3, January 2010 1. 2.3 10 7 4.8 10 4 6 10 5 +9 10 4 1.6 10 3 5 = 1.6 10 2 = 2.3 4.8 107 4 6.9 10 5 = 4.8 103 3 10 5 = 1. 1.6 10-2 2. 7 3. 110,111 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: 1,071 Base 10 = 110,200 Base 3 = 3,060 Base 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 = 2 2 + 2 0 = 4 + 1 = 5 101, 000 = 2 5 + 2 3 = 32 + 8 = 40 11, 111, 111 = 100, 000, 000 1 = 2 8 1 = 256 1 = 255 So our problem becomes 255 5 40 And the answer is 55 = 32 + 16 + 4 + 2 + 1 = 110, 111
Category 3 Meet #3, January 2008 1. Express the base 4 number 3213 4 as a base ten number. 3213 base 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? 23541 7 5635 7 = base 7 1. base 10 2. 3. base 7
Solutions to Category 3 Meet #3, January 2008 1. 231 1. 3213 4 = 3(4 3 )+2(4 2 )+1(4 1 )+3(4 0 ) = 3(64)+2(16)+1(4)+3(1) = 192 + 32 + 4 + 3 = 231 base 10 2. 7 10 or 7.0 10 2. 70000.. 3. 14603 7 3. To subtract 5635 7 from 23541 7 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 14603 7.
Category 3 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 6 )1.4 10 7 ( ) ( 1.1 10 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 1. base nine 2. 3. base four
Solutions to Category 3 Meet #3, January 2006 1. 555 base niine 2. 1.68 10 8 3. 33 base 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 555 base nine. If you do convert 455 base 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 555 base nine. 2. The expression can be solved as follows: ( 1.32 10 6 )1.4 ( 10 7 ) 1.32 1.4 = 106 10 7 1.1 10 9 1.1 10 9 ( ) 1.1 106 7 ( 9) = 1.848 = 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 14x + 20 6 = 224 14x + 14 = 224 14( x + 1)= 224 x + 1 = 16 x = 15 base ten x = 33 base four