Chapter 4 Integers and Number Theory

Transcription

1 Chapte 4 Answes 15 Chapte 4 Intees and Nume Theo Suestions: Althouh it is possile to daw pictues fo the models used in the followin activities, encouae students to use two diffeent coloed squaes to epesent potons and electons and to actuall manipulate the squaes when completin the activities. Activit 1 Answes will va; thee possiilities fo each intee ae iven elow a c. - 2 d a c. 2 d a c a. the nume.. zeo. c. the opposite of the nume. Activit 2 1. Answes ma va. Pai the HEADS with the TAILS and count the nume of coins that cannot e paied. If the emainin coins ae HEADS, move that man units to the iht. If the ae TAILS, move that man units to the left. 2. positive 3. neative 4. No. Since thee ae 6 pennies, the diffeence etween the nume of HEADS and the nume of TAILS must alwas e an even nume. Thus, ou alwas move an even nume of units. 5. If ou toss 3 HEADS and 3 TAILS, ou will end ou tun at the same place ou stated. 6. Answes will va; thee possiilities fo each intee ae iven elow. a. 5H 1T. 7T 4H c. 2H 2T 4H 0T 3T 0H 3H 3T 8H 4T 6T 3H 8H 8T (Continued on next pae.)

2 16 Chapte 4 Intees and Nume Theo Activit 2 Continued 1. a. Each pai means to move 1 unit to the iht and then ack 1 unit to the left, so the net esult is no chane in position.. 2 units to the left c. -2 d Sketches will va. a c. 0 d a. neative. The sum is the neative of the sum of the asolute values of the numes. 4. a. The sum is 0 when the numes ae opposites.. The sum is positive when the nume with the eate asolute value is positive. c. The sum is neative when the nume with the eate asolute value is neative. 5. Find the asolute values of the numes and sutact the smalle asolute value fom the lae one. The sum is the neative of this diffeence if the nume with the eate asolute value is neative; othewise, the sum equals this diffeence. 6. a c d. 0 Activit a. If the asolute value of the positive nume is eate than the asolute value of the neative nume If the asolute value of the two addends is the same c. If the asolute value of the neative nume is eate than the asolute value of the positive nume Find the asolute values of the addends and sutact the smalle asolute value fom the lae one. The sum is the neative of this diffeence if the nume with the lae asolute value is neative; othewise, the sum equals the diffeence The sum of two neative numes is the neative of the sum of thei asolute values EXTENSIONS Answes will va. Two possiilities fo each situation ae iven elow. 1. The pice of a stock inceased 7 points. The followin da it fell 4 points. What was the net chane in the stock s pice? ( ) The tempeatue was -8. B noon, it had isen 3. What was the tempeatue at noon? ( ) 2. The pice of an ounce of old fell $2 one da and $3 the next. What was the net chane in the pice of an ounce of old? A footall team lost 7 ads on one pla and 8 on the next. How man ads did the lose on the two plas? ( )

3 Chapte 4 Answes 17 Activit 4 Suestions: Intoduce this activit eviewin the take awa intepetation of whole nume sutaction and explainin how the intepetation can e extended to the intees. It is impotant to emphasize to students that the enamin used in the sutaction model fo intees is equivalent to addin 0 to the nume. It is also analoous to the enamin used in whole nume sutaction, whee to sutact the units diits, fo example, it is often necessa to ename oowin a B addin 5 potons and 5 electons to a set containin 2 electons. 2. a.. c. Rename Rename Rename d. "Take Awa" -4 e. "Take Awa" +7 f. "Take Awa" -7 Rename Rename "Take Awa" -1 "Take Awa" -8 "Take Awa" a. less than. eate than 4. a c. - 7 d. - 1 e. - 8 f The answes ae the same. If the polem in Execise 2 Pats (a - f) is x, then the coespondin polem in Execise 4 Pats (a - f) is x + ( - ). 6. To sutact an intee, add its opposite. That is, x x a c. -17 d. -22 Activit (Continued on next pae.)

4 18 Chapte 4 Intees and Nume Theo Activit 5 Continued See the second column in the answes fo Execises Fo an two intees a and, a a + (-). EXTENSIONS Answes will va. Two possiilities fo each situation ae iven elow. 1. The aveae Jul tempeatue in Baow, AK, is 39. In Feua, the aveae tempeatue is How much wame is the aveae tempeatue in Jul than in Feua? ( ) The elevation of Mount McKinle is 20,320 ft and the elevation of Death Valle is 282 ft. What is the diffeence in elevation etween Mount McKinle and Death Valle? ( + 20, ) 2. The aveae Apil tempeatue in Baow, AK, is - 2. In Feua, the aveae tempeatue is -20. How much wame is the aveae tempeatue in Apil than in Feua? (-2-20 ) A footall team lost 12 ads on one pla and 8 on the next. How man moe ads did the lose on fist pla than the second? (-8-12) Activit ; a. neative. The sum is the neative of the sum of the asolute values of the numes. 15. a. The sum is positive when the nume with the eate asolute value is positive.. The sum is neative when the nume with the eate asolute value is neative. c. The sum is 0 when the numes ae opposites. 16. Find the asolute values of the numes and sutact the smalle asolute value fom the lae one. The sum is the neative of this diffeence if the nume with the eate asolute value is neative; othewise, the sum equals this diffeence. 17. a The fist numes ae the same, 3, ut the second numes ae opposites, 7 and In each case, the oiinal polem was x and the elated addition polem was x + ( - ). That is, the fist numes ae the same, ut the second numes ae opposites ( - 4)

5 Chapte 4 Answes 19 Activit The poduct of a positive and a neative 6. The poduct of two neative numes nume is the neative of the poduct of is the poduct of thei asolute values. the asolute values of the numes. The quotient of two intees equals the quotient of thei asolute values if oth intees ae positive o oth ae neative; othewise, the quotient of the two intees equals the neative of the quotient of the asolute values. Activit 8 Nume of Squaes Dimensions of the Rectanula Aas , , , 4 1, , , 6 1, 2 3, , , 8 1, 2 4, , 9 1, , 10 1, 2 5, , , 12 1, 2 6, 6 2, 3 4, Total Nume of Aas 1. Nume of Squaes that Poduced: A B C D Onl One Aa Onl Two Aas Moe Than Two Aas An Odd Nume of Aas 1 2, 3, 5, 7, 11,... 4, 6, 8, 9, 10, 12,... 1, 4, 9, a. 8. C c. 1, 2, 3, 4, 6, 8, 12, a. 1, 2, 4, 8, c. C, D (Continued on next pae.)

6 20 Chapte 4 Intees and Nume Theo Activit 8 Continued 4. The nume of factos is equal to the nume of aas. 5. a. The numes in column D ae squae numes; when a squae aa is otated fom hoizontal to vetical, the dimensions emain the same.. 16, 25, a. pime numes. composite numes c. squae numes 7. The squae numes; the all have an odd nume of factos and the squae numes eate than 1 all have moe then 2 factos. One has exactl 1 facto. 8. No 9. a c d a. Yes. The factoizations would all contain the same factos, ut the ode in which the factos ae witten could e diffeent. Activit 9 4 is the fist multiple of 2 to fall thouh the sieve. 9 is the fist multiple of 3 to fall thouh (Continued on next pae.) 100

7 Chapte 4 Answes 21 Activit 9 Continued 1. The ae pime numes ecause the have exactl 2 whole nume factos. When coloin the sieve, the next nume that does not have an cones coloed is pime, and the diamond suoundin the nume is not coloed. 2. The pime numes ae coloed oane and have exactl one cone coloed. 3. Squaes that have one o moe cones coloed and that do not have the diamond suoundin the nume coloed contain composite numes. 1. a It is the squae of the nume ; 999 and 998 ae composite numes since th ae divisile 3 and 2 espectivel. Howeve, 997 is not divisile an pime nume less than 1000 (2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, o 31) so it is a pime nume. Box , 12, 18, 24, 30, 36, 42, 48, 54, 66, 72, 78, 84, 90, the multiples of 6 3. a. Numes coloed with the codes fo 2 and 7 ae multiples of 14.. Numes coloed with the codes fo 2, 3, and 5 ae multiples of 30. Box 4 1. a c Box and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43, 59 and 61, 71 and , 91, 92, 93, 94, 95, Activit The numes on the ame oad ae composite. The ones in the facto list ae pime. 2. a. 2, 3, and 5 ae the pime factos of 60. 4, 6, 10, 12, 15, 20, 30, and 60 ae the composite factos. 1 is onl facto that is neithe pime no composite.. Place two pape clips on 2, one on 3, and one on 5. c. Thee is onl one wa to place the pape clips to cove 60. (Continued on next pae.)

8 22 Chapte 4 Intees and Nume Theo Activit 10 Continued 1. In Step 1, two factos of 45 ae selected to stat the facto tee. The facto 5 is cicled ecause it is a pime. The 9 is not cicled ecause it is composite. 2. In Step 2, each composite facto fom Step 1 is witten as the poduct of two factos. The anches stop at the cicled numes ecause the ae pime and cannot e factoed futhe o Box o Answes will va. One example is iven elow The cicled numes, the pime factos of 120, ae the same. Box 4 1. The pocedues ae the same except that at each step the calculato chooses the smallest pime facto of an emainin composite nume as one of the factos instead of choosin an two factos. 2. a c d Fo an whole nume N, ente the faction N and pess the SIMP and kes. N If the displa shows 1, the nume N is pime. Activit 11 Nume Factoization Nume of Factos The nume of factos is one moe than the exponent. Answes will va. Examples: has 7 factos has 11 factos. In eneal, if p is a pime, then p 6 has 7 factos and p 10 has 11 factos. (Continued on next pae.)

9 Chapte 4 Answes 23 Activit 11 Continued Nume Factoization Nume of Factos To find the nume of factos, add 1 to each of the exponents in the pime factoization, and find the poduct of the sums. [Example: ; nume of factos (2 + 1) (1 + 1) 6] Nume of Factos (3 + 1) (2 + 1) (1 + 1) Factos 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 To find the total nume of factos of a nume, add 1 to each of the exponents in the pime factoization of the nume, and find the poduct of the sums. Answes will va. Example: has 20 factos, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240. Activit is a palindome, that is, its value is unchaned when its diits ae evesed. 1. Answes will va; 1, 4, 9, and 16 ae possiilities. 2. Answes will va; 292, 8338, and 757 ae possiilities. 64 is also a pefect squae. 1. Answes will va; 1, 8, and 27 ae possiilities Box 3 When the diits of 13 ae evesed, the nume fomed, 31, is also a pime nume (101, 131, 151, 181, 191) 2. Answes will va; 17, 37, and 79 ae possiilities. Box 4 1. Answes will va. 2. Nume Ratin Nume Ratin Nume Ratin Nume Ratin if the sum of the diits is eate than 14. (15 pts - pime, 2 pts - 2 factos, 2 pts - odd, 4 pts - sum of diits eate than 14) 4. a. 10 pts fo ein a pefect cue, 7 pts fo ein a pefect squae, 2 pts fo ein odd (the nume could also e even and scoe 4 pts), and 1 pt fo each of the followin factos: 3 1, n, n, n 3, and( n ) 2. (Thee ma also e othe factos which would make the inteest atin even hihe.) o c Inteest Ratin 37

10 24 Chapte 4 Intees and Nume Theo Activit A 4 8 ectanle can e tiled with 1 1, 2 2, and 4 4 squaes as shown elow. The ectanle cannot e tiled with 3 3 squaes since the lenth of a side of one squae is less than the 4 unit width of the ectanle and two squaes side--side have a lenth of 6 which is eate than the 4 unit width No othe squaes can e used to tile a 4 8 ectanle since the lenth of a side of the squae would e eate than the 4 unit width of the ectanle. Dimensions of Lenth Dimensions of Common Squaes of Side of Pime Factoization Rectanle Factos That Ma Be Laest width lenth width lenth Used Squae , 2 2, , , 2 2, 6 3 3, , 2 2, , 4 4, 6 6, The lenth of a side of the laest squae that can e used to tile a ectanle equals the poduct of the common factos. 1. a. 1, 2, 3, 6, 9, 18. 1, 2, 3, 5, 6, 10, 15, 30 c. 1, 2, 3, 6 d a. 1, 2, 3, 4, 6, 8, 12, 24. 1, 2, 3, 4, 6, 9, 12, 18, 36 c. 1, 2, 3, 4, 6, 12 d The GCD is the same as the lenth of a side of the laest squae that can e used to tile the ectanle. 4. The lenth of the side of the squae is GCD(m,n) which is the poduct of the common pime factos of m and n. 1. a. GCD(105,126) 21. GCD(38,95) 19 c. GCD(420,714) 42 d. GCD(924,770) Like the pocedue in Execise 4, the calculato simplifies the faction findin the common pime factos of the two numes and dividin oth the numeato and denominato them.

11 Chapte 4 Answes 25 Activit 14 Heiht Base No. of Hits No. of Squaes Final Pocket D C B Nume of Hits (Base + Heiht) GCD(Base, Heiht) Nume of Squaes LCM(Base, Heiht) EXTENSION J Heiht GCD(Base, Heiht) If J is odd and K is even, Final Pocket is B. K Base GCD(Base, Heiht) If J is odd and K is odd, Final Pocket is C. If J is even and K is odd, Final Pocket is D. The all neve oes in A.