Black Problems - Prime Factorization, Greatest Common Factor and Simplifying Fractions
1. A natural number n , such that n > 1, can’t be written as the sum of two more consecutive odd numbers if and only if n is prime or n is twice an odd number. Of the ten natural numbers 20 through 29, how many can’t be written as the sum of two or more consecutive odd numbers?
2. Count prime dates. A prime day has a month and day which are both prime. Thus 5/13 is a prime date. How many prime dates are there in the year 2000? [Dec. 1998 (3)]
3. What is the positive difference between the greatest and least prime factors of 2000?
4. What is the greatest prime factor of 55
100+ 55
101+ 55
102?
5. What is the least whole number value of x such that f ( x ) = x
2+ x + 11 is not prime?
6. The 26 letters of the alphabet are assigned prime number values in consecutive order beginning with 2. The product of the values of the letters of a common math term is 595,034. What is the term?
7. What is the only prime number which is the sum of four consecutive prime numbers?
8. What is the smallest prime factor of 1821?
9. What is the least positive fraction whose numerator is two less than a perfect square and whose denominator is one more than the same perfect square?
10. Between 3:00 p.m. and 4:00 p.m., for what fractional part of the hour does the digit
“2” appear on a 12-hour digital clock that shows hours and minutes? Express your answer as a common fraction.
11. Simplify:
a
2-5 3 a am -5 +6 m m
2+
a -2 1 m - a -3 2 m .
12. Think about all of the positive common (reduced) fractions with a denominator of 11 or less. How many of these common fractions have a value less than
117?
13. Supposed we replace each x in the expression
x - 1x
+ 1with the expression
x - 1x
+ 1. What is the value of the resulting expression when x =
54
? Express your answer as a
common fraction.
14. ABCD is a rectangle, AB = 6, BC = 4, EFGH is a parallelogram, AE/BE = 2/1, and BF/FC = 1/3. What is the ratio of the area of parallelogram EFGH to the area of rectangle ABCD? Express your answer as a common fraction.
A E B
D G C H
F
15. Both 5 and 29 are prime numbers that can be written as sums of squares: 5 = 1
2+ 2
2and 29 = 2
2+ 5
2. Which of the following numbers has the same properties: 137, 71, 153, 23, 169?
Black Solutions
1. The primes from 20 to 29 are 23 and 29. The numbers that are twice an odd number are 22 and 26.
These 4 natural numbers can’t be written as the sum of two or more consecutive odd numbers.
2. Count prime dates. Fifty-three. Prime months are 2, 3, 5, 7, and 11; prime days are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. If all prime months had 31 days, there would be 55 prime dates. But February in 2000 has 29, and November has 30. So there are 53 prime dates.
3. The prime factorization of 2000 is 24 x 53. The greatest prime factor is 5, and the least prime factor is 2. The difference between these two numbers is 5 – 2 = 3.
4. Factoring 55100 + 55101 + 55102 gives 55100(1 + 55 + 552) = 55100(3081). Further factorization gives 5100 x 11100 x (3 x 13 x 79). The greatest prime factor is 79.
5. Value of the function can be checked with a chart.
x x2 + x + 11 0
1 2 3 4 . . . 10
11 13 17 23 31 . . . 121
When x = 10, the value of the function is x2 + x + 11 = 102 + 10 + 11 = 121. This is the first value of the function which is not prime, so the answer is 11.
6. Since the product consists of values that are primes, find the prime factorization of 595,034. When factoring only try prime values, because if a prime doesn’t work then no multiple of that prime will work. The number is even, so quite obviously 2 is a factor: 595,034 = 2 x 297,517. Trying to find factors of 297,517 is a bit more difficult; 3 doesn’t work, 5 doesn’t work, 7 doesn’t work. The first to work is 11: 297,517 = 11 x 27,047. Using a calculator then yields that 27,047 = 17 x 37 x 43. The prime factors are 2, 11, 17, 37 and 43, and these primes correspond to the letters a, e, g, l, and n, which can be rearranged to spell angle.
7. Every prime number except 2 is odd. The sum of any four odd numbers is even. Thus, the only way to get an odd sum for four primes is if one of the primes is 2. The only set of consecutive primes that
contains 2 is the set {2, 3, 5, 7}. The sum of these four primes is 17. Thus, 17 is the only prime that can be written as the sum of four consecutive primes.
8. 3
9. The perfect squares around which our fraction will be built are 1, 4, 9, 16, etc. We cannot use 1 since the numerator will become negative when we subtract 2. Let’s try 4. The numerator is two less, which is 4 – 2 = 2, and the denominator is one more, which is 4 + 1 = 5. The fraction is 2/5. To be sure, we check that the next possibility, 7/10, is clearly more than 2/5, as is 14/17, and the rest, which get closer and closer to a value of 1. In each case out numerator is three less than our denominator. In terms of a pizza, for instance, the smaller the denominator (or number of total pizza pieces), the more of an impact it is to decrease the numerator by three (take away three pizza pieces), so we want the smallest possible denominator.
10. The digit 2 will appear on the digital clock for a full ten minutes from 3:20 through 3:29 and for five more full minutes at 3:02, 3:12, 3:32, 3:42, and 3:52. That’s 15 minutes in all, which is 1/4 of the hour.
11.
a
2-5 3 a am -5 +6 m m
2+
a -2 1 m - a -3 2 m = ( a - 2 3 a m -5 )( a m - 3 m ) +
a - 2 1 m - a - 3 2 m
= 3 a + 5 m ( a + - 2 a - 3 m )( a m - 3 - 2( m ) a - 2 m )
= 3 a - 5 ( m a - 2 + a m - 3 )( a m - 3 - 2 m a ) + 4 m
= ( a - 2 2 a m - 4 )( a m - 3 m )
= ( a - 2 2( a m )( - 2 a - 3 m ) m )
= a - 3 2 m
12. A positive fraction can be represented as a lattice point on a grid using the ordered pair (denominator, numerator). One fraction is less than another fraction if the segment joining it to the origin (0, 0) is below the segment joining the other fraction to the origin. In Figure 1 we have drawn the segment connecting
117 to the origin with a solid line. Every other common fraction with a denominator of 11 will fall along the line x = 11. The segments representing the positive fractions with a denominator of 11 and a value less than
117 are drawn with dotted lines. We can see
there are only six dotted lines, so there are six positive fractions with a denominator of 11 that are less than
117 . We can then go to a denominator of 10 and see how many of these fractions work. In other words, how many lattice points are along the line x = 10 and below the segment representing
117 . Notice that when we eventually draw in the corresponding segment for the fraction
51 , if will fall along the exact same segment that was drawn for
102 , so we’ll know not to count it again. This is how we will ensure we are counting only common fractions.
We can see from Figure 2 that this is going to get messy and difficult to count, so it might not be the best method. However, without drawing in all of the lines, we can get good idea from the lattice points about which fractions are less than
117 . Careful graphing and counting yields an answer of 26. In Figure 3 we see there are 36 lattice points below the segment representing
117 (and above the x-axis) but the 10 open- circle lattice points do not represent common fractions. The answer is
36 – 10 = 26 fractions.
7
11 Figure 1
7
11 Figure 2
7
11 Figure 3
13. Substituting the value 4/5 into expression (x + 1)/(x – 1), we get (4/5 + 1)(4/5 – 1) = (9/5)/(-1/5) = -9.
Now we substitute this value again into the expression (x + 1)/(x -1), which gives us (-9 + 1)/(-9 – 1) = (- 8)/(-10) = 4/5.
14. The area of rectangle ABCD is 6 x 4 = 24 square units. Triangles AEH and CFG together make a rectangle that is 3 x 4 = 12 square units. Likewise, triangles DGH and BEF together make a rectangle that is 1 x 2 = 2 square units. Subtracting 12 and 2 from 24, we find that parallelogram EFGH has an area of 10 square units. The ratio of the area of parallelogram EFGH to rectangle ABCD is
2410
= 125 . 15. Notice that 137 is prime. We must only look at values that are less than
137
. By trial and error wefind that 137 = 121 + 16 = 112 + 42.
