More Related Content More from Mosab Qasem (20) 4. ∂JÉeƒ∏©e ÈàNG äÉHÉLEG øjQɪàdGh á∏Ä°SC’G πM
(13 - 12) äÉëØ°üdG ¤hC’G IóMƒ∏d
áHÉLE’G áHÉLE’G õeQ Iô≤ØdG ºbQ áHÉLE’G áHÉLE’G õeQ Iô≤ØdG ºbQ
4 Ü 9 15 `L 1
45^902 Ü 10 29 `L 2
0^2178 CG 11 1 `L 3
4
8 CG 12 9 Ü 4
2
378
Ü 13 3 3
4
Ü 5
21 `L 14 2^5 Ü 6
16
25 Ü 15 7 Ü 7
15 `L 8
238
5. (1 - 1) πFÉ°ùeh øjQÉ“
(19) áëØ°U
2 = 2 ( CG (1
1
18- 36
5 = 10 - = 3^6- (Ü
4- 8
5 = 10 - = 0^8- (`L
18 3+ 5 * 3 3
5 = 5 =3 5 ( O
5 1+ 2 * 2 1
2 = 2 = 2 2 ( CG (2
=( 2 )- = 1 1 - (Ü
3- 1+1*2
2 2
6 á≤∏£eh 6- á°Sƒµ©e 6 ( CG (3
18 18 18
17 á≤∏£eh 17 á°Sƒµ©e 17 - = 1 7 - (Ü
10 10 10 10
2 á≤∏£eh 2 - á°Sƒµ©e 2 = 0^2 (`L
10 10 10
12 á≤∏£eh 15 áHƒ∏≤e 12 ( CG (4
15 12 15
24 á≤∏£eh 2- = 48- = áHƒ∏≤e 24 (Ü
48 24 48-
19 á≤∏£eh 7 = áHƒ∏≤e 19 = 6 _ 114 = 72 + 1 * 42 =1 72 (`L
7 19 7 6 _ 42 42 42
. kÉë«ë°U Gk OóY ¢ù«d ¬æµd »Ñ°ùf OóY ƒg 1 kÓãªa ,CÉ£N IQÉÑY ( CG (5
2
.áë«ë°U IQÉÑY (Ü
1 40 _ 40 40
3 = 40 _ 120 = 120 (6
239
6. (2 - 1) πFÉ°ùeh øjQÉ“
(26) áëØ°U
o p o
:áë«ë°U á«JB’G äGQÉÑ©dG ípÑ°üJo å«r ëHp ™HôªdG »a = , > , < õesôdG ™°V (1
k n n o o n pq o nr p n
7 7 3 - 5 -
13 - > 11 - ( Ü 3 > 3 ( CG
3 1 - = 7 - ( O 3 - 1
< 2 - (`L
2 2 5
o n n n rn p n
:ÉvjóYÉ°üJn á«JB’G OGóYC’G ÖuJQ ( 2
.ájhÉ°ùàe áë«ë°üdG OGóYC’G ¿C’ Qƒ°ùµdG ¿QÉ≤f 1 2 - , 1 1 - , 1 2 - ( CG
3 3 5
2 - > 2 - ¿EÉa 5 * 2- < 3 * 2- ¿C’h 1- > 2- ¿EÉa 5 * 1- > 3 * 2- ¿CG ɪH
5 3 3 5
1 2 2
1 3 - > 1 5 - > 1 3 - ¿PEG
2 1- , 1 1- , 1 3- (Ü
3 2 5
1 1- > 1 3- Gòd 5 * 1- > 2 * 3- ¿CG ɪHh ,1 1- ™e 1 3- ¿QÉ≤f ∂dòd OóY ô¨°UCG ƒg 2 1-
3 5 2 5 3
1 1 - > 1 3 - > 2 1 - ¿PEG
2 5 3
1 2 - , 1 1 - , 4 = 28 (`L
5 3 3 21
1 2 - ™e 1 1 - ¿QÉ≤f ∂dòd ÖLƒe ¬fC’ ÉgôÑcCG ƒg 4 = 28
5 3 3 21
1 1 - > 1 2 - Gòd 3 * 2- < 5 * 1- ¿CG ɪH
3 5
28 1 2
21 > 1 3 - > 1 5 - ¿PEG
1 4 - = 9 - , 1 17 - , 1 12 - ( O
5 5 20
7
Qƒ°ùµdG ¿QÉ≤f ájhÉ°ùàe áë«ë°üdG OGóYC’G ¿CG ɪH
¿PEG 12 * 4- < 5 * 7- h 12 * 17- < 20 * 7- h 20 * 4- > 5 * 17- ¿CG ɪHh
1 12 > 1 4- > 9 - >1 17 -
7
5 5 20
o n n n r p n
:Év«dRÉæJn á«JB’G OGóYC’G ÖuJQ (3
19 - , 5 , 5 - , 2 ( CG
24 8 6 3
¿CG ɪHh 5 - < 19 - Gòd 24 * 5- < 6 * 19- ¿CG ɪHh , 5 < 2 Gòd 3 * 5 < 8 * 2 ¿CG ɪH
6 24 8 3
. 5 - < 19 - < 5 < 2 ¿PEG áÑdÉ°ùdG øe ôÑcCG áÑLƒªdG OGóYC’G
6 24 8 3
240
7. (2 - 1) πFÉ°ùeh øjQÉ“
(26) áëØ°U
1 , 2 , 3 , 3 , 2 , 1 - (Ü
2 3 - 4 - 4 3 2
¿PEG 9 > 8 ¿CG ɪH 3 ™e 2 h 2 > 1 ¿PEG 4 > 3 ¿CG ɪH , 2 ™e 1 …CG áÑLƒªdG Qƒ°ùµdG ¿QÉ≤f
4 3 3 2 3 2
¿ƒµj É¡Ñ«JôJ ¿EÉa IQÉ°TE’G ±ÓàNG ™e É¡°ùØf áÑLƒªdG Qƒ°ùµdG »g áÑdÉ°ùdG Qƒ°ùµdG ¿CG ɪH 1 < 2 < 3
2 3 4
¿PEG , 3 - < 2 - < 1 -…CG áÑLƒªdG É¡Jô«¶f ¢ùµY
4 3 2
3 -< 2 -< 1 -< 1 < 2 < 3
4 3 2 2 3 4
1 1 3 1
6 - Oó©dG πãe 3 - h 0 øjOó©dG ø«H Iô«ãc OGóYCG ∑Éægh , 1 4 = Ü 3 - = CG ( 4
4+2*3 1 + 3
5 = 10 = 1 * 10 = 8 2 4
8 16 2 8 2 = 2 (5
?ôÑn rcC’G Oó©dG ƒg OGóYnC’G √p òg øe w…CGn . CG < `L , Ü < CG , Ü < `L ( 6
o
n onn n o p n r p r p
Ü < CG < `L ¿C’ `L
241
8. (3 - 1) πFÉ°ùeh øjQÉ“
(34) áëØ°U
¬àæe 2^75- = 2 3 - ( O
4 …QhO 1^3= 4 ( `L
3 ¬àæe 1^5= 1 1 (Ü
2 ¬àæe 0^6= 3 ( CG (1
5
4- 1
…QhO 0^571428- = 7 ( h …QhO 0^142857 = 7 ( `g
87936 Oó©dG 6^287936 ( O 35 Oó©dG 2^35 ( `L 421 Oó©dG 3^421 ( Ü 7 Oó©dG 0^7 ( CG (2
254 1000 * 0^254 -
1000 -= 1000 * 1 = 0^254- ( CG (3
2137 100 * 21^37
100 = 100 * 1 = 21^37 (Ü
6009 1000 * 6^009
1000 = 1000 * 1 = 6^009 ( `L
132619 131 7 7
9990 = 13^2751 ( O 990 = 0^132 ( `L 90 = 0^07 (Ü 3 = 2^3 ( CG (4
9 1
0^33 = 0^3 ( O 5 -
< 1^7- (`L
5
< 0^3 (Ü 2^3 > 0^23 ( CG (5
1^4- , 0^2 , 1^2- ( CG (6
¿PEG 1^4- < 1^2- ¿ƒµ«a , 1^4- , 1^2- ø«ÑdÉ°ùdG øjOó©dG ¿QÉ≤fh ôÑcC’G ƒg 0^2 ÖLƒªdG Oó©dG
1^4- < 1^2- < 0^2
0^1 , 0^4 , 0^09- , 0^1 - (Ü
¿PEG áÑdÉ°ùdG øe ôÑcCG áÑLƒªdG OGóYC’Gh 0^9- < 0^1- ∂dòch 0^1 < 1^4 ¿CG ɪH
0^1 - < 0^09 - < 0^1 < 0^4
2^45- ,0^6 , 1^85- ( CG (7
0^6 > 1^85 - > 2^45- ¿PEG 2^45- < 1^85- kÉ°†jCGh ÖLƒe ¬fC’ OGóYC’G ôÑcCG ƒg 0^6
4 = 0^4 , 1 , 6 Qƒ°ùµdG ¿QÉ≤f É¡©«ªL OGóYC’G »a …hÉ°ùàe í«ë°üdG Oó©dG ¿C’ 1^4 , 1 1 , 1 6 (Ü
10 3 7 3 7
6 > 0^4 > 1 ¿PEG 1 < 4 ¿EÉa 10 * 1 < 3 * 4 ¿CG ɪHh 4 < 6 ¿EÉa 4 * 7 < 10 * 6 ¿CG ɪHh
7 3 3 10 10 7
4
…QhO ô°ùµdG , 1^571428 = 1 7 (8
242
9. (4 - 1) πFÉ°ùeh øjQÉ“
(40 - 39) áëØ°U
10 * 3^6 = 10000 * 36000 = 36000 ( CG (1
4
10000
9
10 * 9 = 100000000 * 9000000000 = 9000000000 ( Ü
1000000000
8
10 * 1^25 = 100000000 * 125000000 = 125000000 ( `L
100000000
7
10 * 3^45= 610 × 34^50 ( O
4500000 o p o on n
᪰ùf 610 * 5^4 = 1000000 * 1000000 = ᪰ùfn 5400000 u¿OQC’G ¿Éµ°S OóY ( CG (2
r
ºc 210 * 4^2 = 100 * 420 = ºc420 áÑ≤©dGh ¿ÉªY ø«Hn áaÉ°ùªdG (Ü
n q n n on n
100
ç/ºc 510 * 3 = 100000 * 300000 = á«fÉãq dG »a ôàe ƒ∏«c ∞dCG 300 Ap ƒs°†dG áYô°S ( `L
100000
p on o
9
10 * 1^2 = 1000000000 * 1^2 = QÉ«∏e 1^2 ºdÉ©dG »a ø«ª∏°ùªdG OóY ( O
p n o on n
12
10 * 4^32 = 1800000 × 2400000 ( CG (3
10 * 1^20009 = 1200000000000 + 90000000 ( Ü
12
10 * 2^9999975 = 25000000 – 30000000000000 ( `L
13
10 * 7^2 = 90000000 × 80000000 ( O
15
(4
≈dhC’G á°TÉ°ûdG (5
9
10 *1^4 = 400000000 = 100 * 14000000
á«fÉãdG á°TÉ°ûdG
8
10 * 5^8 = 20000000 + 100 * 560000000
GQÉæjO 610 * 3 = 3000000 = 120 * 1000 * 25 (6
k
243
10. (5 - 1) πFÉ°ùeh øjQÉ“
(47 , 46) áëØ°U
1 1 1 1 1
119^75 = 134 2 - 254 4 ( `L 6 = 10 2 + 4 2 - ( Ü 9^5- = 3 2 - 6- ( CG (1
3 = 9 _ 27 = 3 *5 + 6 * 2 = 5 + 2
2 9 _ 18 3*6 6 3 ( CG (2
7 1 = 86 = 53 + 33 = 53 + 11 = 5+4*12 + 3 + 2 * 4 = 4 5 + 2 3 ( Ü
6 12 12 12 12 4 12 4 12 4
17- = 8- + 9- = 4*2- +3 * 3- = 2- + 3- ( `L
12 12 3*4 3 4
5 = 3 * 1- + 2 * 4 = 1- + 4 ( O
6 2*3 2 3
4_12 12 ( 2*6 ) ( 2 6 )
7 2 - = 23 - = 4_92 - = 54 + 38 - = 6*9+2*19 - = 9 + 19 - = 4 1 - + 3 1 - ( `g
3 3 2 6
1 1 - = 9 - = 2 - + 7 - = 1- - + 7 - ( h
8 8 8 8 4 8
1 1 - = 5- = 8- + 3 = 8 - + 3 = 8 - 3 ∫hC’G πµ°ûdG (3
4 4 4 4 4 4 4
1
( 1 1
)
4 = 3 + 1 = (1 + 2) + 2 + 2 = 1 2 + 2
1 »fÉãdG πµ°ûdG
2
5 5 (
1 4 -= 9 -= 6 + 3 -= 6 -+ 3
5 5 )5 5 - ådÉãdG πµ°ûdG
1 8 - = 23 - = 4 _ 92 - = 10 * 5 + 6 * 7 - = 5- + 7- = 5 - 7- ( CG (4
15 15 4 _ 60 10 * 6 6 10 6 10
3- = 6- = 2 + 8- = 2 * 1 + 8 * 1- = 1 + 1- = 1- - 1-
8 16 16 8*2 8 2 8 2 (Ü
2 1 3
( 1 3
)
1 5 = (1 - 2) 5 - 5 = 1 5 - 2 5 ( `L
5 2+3 1 3
( 1
)
3
12 8 - = 12 - + 8 - = (5 - 7-) + 4 + 8 - = 5 4 - 7 8 - ( O
3 3 3 3 3
2 7 - = 2- + 7 - = (3 + 5-) + 7 - = 3 + 5 7 - = 3- - 5 7 - ( `g
3 3 3
9 7 =5 7 +4=5 7 --4 ( h
(1 1
15 = 14 + 1 = (2 + 12) + 2 + 2 ) = 2 1 +12 1 = 2 1 - - 12 1
2 2 2 2 (5
244
11. (6 - 1) πFÉ°ùeh øjQÉ“
(53) áëØ°U
16= 48 = 8 * 6 =2 2 * 6 (Ü 4 = 12 = 6 * 2 = 6 * 2 ( CG (1
3 3 1 3 7 21 7 * 3 7 3
1 - = 25 - = 5- * 5 =0^5- * 5 ( O 45 = 180 = 36 * 5 = 36 * 5 ( `L
6 150 10 15 15 2 8 8 8
72 = 27- * 8- = 5 2 - * 2 2 - 7- = 7 * 5- = 7 * 5-
( h ( `g
5 5 3 5 3 18 10 * 9 10 9
1 = 5 = 1- * 5- = 1- * 0^5- ( ì 3 = 42 = 6- * 7- = 6- *1 3- ( R
12 60 6 10 6 2 28 7 4 7 4
1144 = 44 * 26 = 8 4 *3 5
35 5 7 5 7 (•
4 < 2 _ 4 (Ü 9
> 1 * 4 ( CG (2
3 4 9
9= 3 * 6 ( O 3
> 4 _ 3 (`L
2 5 5
21 = 42 = 2 * 21= 2 1 _ 5 1 ( Ü 18 = 5*36 = 5 * 36 = 2 _ 7 1 ( CG (3
10 20 5 4 2 4 2*5 2 5 5 5
7 - = 14- = 7 * 2- = 5 3 _2- ( O 5 - = 25- = 1 * 25 - = 5 _ 3 4- ( `L
19 38 38 7 7 35 5 7 7
98 = 7 * 14 = 2 1 _14 ( h 28 = 84 = 6 * 14 = 4 1- _1 5 - ( `g
15 15 7 75 225 25 9 6 9
1 = 1 * 1 =4_ 1
16 4 4 4 ( R
5 28 68 1 3 5
Ω 136 = 2 * 5 * 7 = 2 2 * 5 5 * 9 7 = ´ÉØJQ’G * ¢Vô©dG * ∫ƒ£dG = ¿GõîdG ºéM (4
3
7
3
Ω 119 = 8 * 136 = áÑ©µªdG QÉàeC’ÉH AɪdG ºéM
45 5 9 1 9
áYÉ°S 4^5 = 10 = 1 * 10 = 5 _ 10 (5
245
12. (7 - 1) πFÉ°ùeh øjQÉ“
(59 - 58) áëØ°U
(Üô°†dG ºK ø«°Sƒ≤dG πNGO ìô£dG) ( CG (1
19 = 19 * 3 = 2 * 2 + 3 * 5 * 3 = 2- - 5 * 3 = 2- - 2 1 * 1 1
4 6 2 2*3 2 3 2 2 3 2 2
(™ªédG ºK Üô°†dG) ( Ü
4*160+9 *13- 160 13- 10 16 13-
= + = * + =3 1 *51 +31 -
9*4 9 4 3 3 4 3 3 4
523 = 640+117- =
36 36
(Üô°†dG ºK ø«°Sƒ≤dG πNGO ™ªédG) ( `L
85- = 10- * 17 = 31- + 7 * 17 = 5 1 - + 3 1 * 4 1
12 6 4 6 2 4 6 2 4
7 - 8 = 5*1-+4*3 - 8 = 1- + 3- 2 2 = (´ + ¢U) – ¢S ( CG (2
20 3 5*4 3 4 5 3
139 = 3*7-20*8 =
60 20*3
15*1-+4*31 = 1-+ 31 = 1- + 3*3-5*8 = 1- + 3 - 8 = ´ + ¢U – ¢S ( Ü
4*15 4 15 4 5 *3 4 5 3
109 = 15-124 =
60 60
181 15-+4 *49 1- 49 1- 3*3+5*8 1- 3 8
60 = 4*15 = 4 + 15 = 4 + 3*5 = 4 - 5 + 3 = ´ – ¢U + ¢S ( `L
14 = 56 = 7 * 8 = 7 * 8 = (´ + ¢U) × ¢S ( O
15 60 20 * 3 20 3
89 = 3*3+10*8 = 3- - 8 = 1- * 3 - 8 = (´2 × ¢U) – ¢S ( `g
30 10*3 10 3 2 5 3
44 = 88 = 11 * 8 = 5*1+2*3 * 8 = 1- - 3 * 8
15 30 10 3 2*5 3 2 5 3 = (´2 – ¢U) × ¢S ( h
0^21- = 21- = 21 * 2- = 1 + 2 * 2- = 1 + 0^4 * 2- ( CG (3
100 40 5 8 5 5 8 5
0^04- = 0^2- * 0^2 = (0^5 - + 0^3) * 0^2 = 1- + 0^3 * 0^2 ( Ü
2
6^24 = 2 - 8^24 = 125 * 8 - 8^24 = 1^25 * 1 3 - 8^24 ( `L
100 5 5
45^535 - = 38^385 - + 7^15 - = 8^53- * 4^5 + 7^15- = 8^53- * 4 1 + 7^15- ( O
2
246
13. (7 - 1) πFÉ°ùeh øjQÉ“
(59 , 58) áëØ°U
10 16 5 3 1 5
21 = 21 * 8 = 7 + 3 * 8 ( CG (4
35 7 5 1 2 5
12 =3* 6 * 6 = 3 * 2 + 3 * 6 (Ü
5 = 15 = 5 + 5 = 5 + 3 * 1 - 4 * 2 = 5 + 1 - 2 ( CG (5
4 12 6 12 6 4*3 6 4 3
4 = 4 5 + 5- = 4 5 + 52 - 42 = 4 5 + 13 - 7 = 4 5 + 2 1 - 1 3 ( Ü
12 12 12 24 12 6 4 12 6 4
91 = 27 + 64 = 3 + 8 = 3 + 2- * 4-
72 72 8 9 8 3 3 ( `L
9 = 7 - 16 = 7- + 4 = 7 * 1- + 4
20 20 20 5 10 2 5 ( O
71- = 100 + 171- = 4*25+3* 57- 25 57- = 25 9 19- =8 1 4 1 3 1-
12 12 12 = 3 + 4 3 + 2 * 6 3 + 2 * 6 ( `g
13 - = 13 - * 5 = 2 3 - * 1 5 - + 2 1
12 5 12 5 6 4 ( h
1^5 = 0^25 - 0^75 - 2^5 = ( R
2 = 0^8 _ 1^6 = 0^8 _ 2 _ 3^2 = ( ì
™ªédG πÑb Üô°†dG ≈dhC’Gh Üô°†dG πÑb ™ªédG á«∏ªY AGôLEG ƒg CÉ£îdG (6
10 1 = 5 1 + 5 = 21 + 5 = 3 * 7 + 5 = 1 1 * 3 1 + 5
4 4 4 2 2 2 2
247
14. (8 - 1) πFÉ°ùeh øjQÉ“
(67) áëØ°U
33 »fÉãdG ™HôªdGh , 1 ∫hC’G ™HôªdG ( Ü 24 »fÉãdG ™HôªdGh , 8 ∫hC’G ™HôªdG ( CG (1
(Ü ( CG (2
pr n p o o r
:ø«Ñ°SÉæJ ≈∏Y ∫ƒ°üë∏dp »JCÉjn Ée »a äÉZGôØdG πª rcCG (3
p n p p
8 = »fÉãdG ™HôªdG , 28 = ∫hC’G ™HôªdG ( CG
28 = ådÉãdG ™HôªdG , 7 = »fÉãdG ™HôªdG , 36 = ∫hC’G ™HôªdG ( Ü
24 = 12 ∫hC’G (4
10- 5-
10- = 24 »fÉãdG
5- 12
14 = 7 24 + 10- = 12 + 5- ådÉãdG
24 12 24 12
34- = 17- 24 - 10- = 12 - 5- ™HGôdG
24 12 24 12
11 - 8 = ¢U - ¢U + ¢S (5
11 ¢U
3 - = ¢S
11 ¢U
4 = 4 - ¢S
3 4 ( CG (6
28 = ¢S ¿PEG 7 = ¢S É¡æeh 3 + 4 = 4 + 4 - ¢S
3 3 4 3 4
2 = 4
15 ¢U 5 ( Ü
¢U
6 = ¢U ¿PEG 2 = É¡æeh 4 = ¢U 5
3 2 15
8- = 2 + ∫
9 2- ( `L
2- = ∫ ¿PEG 1 = ∫ É¡æeh 9 + 8- = 2- + 2 + ∫
9 9 2- 9 2-
1 = ¢S ¿PEG 2-3 = 1 - 1 + ¢S É¡æeh 3 = 1 + ¢S É¡æeh 1 = 1 + ¢S ( O
2 2 1 2 1 2 3
248
15. (9 - 1) πFÉ°ùeh øjQÉ“
(72) áëØ°U
n pq o p
.ø«ª∏©ªdG ánaô oZ Gó©Ho ( CG (1
Ω10 = º°S 1000 = º°S 200 * 5 = »≤«≤ëdG ø«ª∏©ªdG áaôZ ∫ƒW
Ω 8 = º°S 800 = º°S 200 * 4 = »≤«≤ëdG ø«ª∏©ªdG áaôZ ¢VôY
»°SGQuódG u∞s°üdG ¢VôY ( Ü
o
Ω 7^2 = º°S 720 = º°S 200 * 3^6 = »≤«≤ëdG ∞°üdG áaôZ ¢VôY
1 8
400000 = 3200000 = º°SôdG ¢SÉ«≤e (2
360000
º°S 18 = Ω 0^18 = 2000000 = á£jôîdG »a ᪫≤à°ùªdG á©£≤dG ∫ƒW (3
1
Æ 7 = 35 * 5 = ÖgòdG ¿Rh (4
4
Æ 28 = 35 * 5 = á°†ØdG ¿Rh
10 3
mQÉæjO 2000 = 3 * 600 = ≠∏ѪdG É¡æeh 600 = ≠∏ѪdG * 10 (5
6 : 2 : 3 Ö°ùædG íÑ°üJ äÉeÉ≤ª∏d ∑ôà°ûªdG ∞YÉ°†ªdG Üô°†dÉH (6
3
QÉæjO 900 = 3300 * 11 = óªMCG Ö«°üf
2
QÉæjO 600 = 3300 * 11 = »fÉg Ö«°üf
6
QÉæjO 1800 = 3300 * 11 = »ª¡a Ö«°üf
Ω 4 = º°S 400 = º°S 50 * 8 = á∏aÉë∏d »≤«≤ëdG ∫ƒ£dG ( 7
249
16. á©LGôe
(74 , 73) áëØ°U
11 - = 3 + 2 * 4 - = 2 3- ( Ü 6- = 6- ( CG (1
4 4 4 1
0 33-
r p
1 = ôØ°U ( O 100
= 0^33- ( `L
0^4 > 0^1- > 3- > 0^7- > 5- (2
5 3
3 = 18 = 9 * 2 = 2 * 2 + 5 * 2 = 2 + 1 * 2 ( CG (3
5 30 10 3 5*2 3 5 2 3
7^6 = 2 + 5^6 = 4 + 5^6 = 4 * 1 + 5^6 ( Ü
2 2
11- > 3^2- ( Ü 3- > 4- ( CG (4
15 8 7
0^4 = 2 2 > 2
( O ( `L
5 9 11
4 1 3 ( CG (5
= +
5 5 5
5 3-8 3 8
= = -
11 11 11 11 ( Ü
24 6 * 4 6 4 ( `L
= = *
35 5 * 7 7 5
9 = 3 * 3 = 1 _ 3 (O
5 1 5 3 5
3- = 6 + 9- ( `g
10 10 10
10 = 5 - 15 = 5 - 5 ( h
21 21 21 21 7
3 6 3*2 3 2-
5 ( R
-= -= -= *
20 40 8*5 8
3 = 15 = 3 * 5 = 5 _ 5 ( ì
3 5 3
7 37 5*5+6*2 5 2
6 =5 =5 =3 +2 (•
30 30 6*5 6 5
2 8 = 38 = 25 - 63 = 5*5-3*21 = 5 - 21 = 1 2 - 4 1 ( …
15 15 15 3*5 3 5 3 5
15^3 = 153 = 9*17 = 9 * 17 = 4 1 * 3 2 ( ∑
10 2*5 2 5 2 5
250
17. á©LGôe
(74 - 73) áëØ°U
10 2 = 32 = 8 * 28 = 7 _ 9 1 ( ∫
3 3 7 3 8 3
4 19 = 3 79 = 3 + 15 + 24 + 40 = (2 + 1) + 1 + 2 + 2 = 1 + 12 + 1 2 ( Ω
60 60 60 60 60 4 5 3 4 5 3
5 17 =5 4 * 2 - 5 * 5 =3 2 - 8 5 = 17 - 9 1 ( ¿
20 20 5 4 5 4
10 2 = 92 = 8- *23- = 2 2- * 3 5- ( ¢S
9 9 3 6 3 6
19- = 1 * 19- = 5 5-
_2 (´
35 5 7 7
Gôàd 111 = (24 - 135) + 1 - 1 = 24 1 - 135 1 ( 6
k 2 2 2 2
º°S 0^04 = 1 = ¢S 3 = ¢S 1^5 = ¢S 1^5 = 75 ( 7
25 75 75 2 ¢S 2
Ωóîà°ùªdG ¢ù∏WC’G ≈∏Y áHÉLE’G óªà©J ( 8
1400 á«fÉãdG Ö«°üf + ≈dhC’G Ö«°üf á«fÉãdG Ö«°üf ≈dhC’G Ö«°üf
200 = = = = ( 9
7 5+2 5 2
≈dhC’G Ö«°üf
QÉæjO 400 = 200 * 2 = ≈dhC’G Ö«°üf É¡æeh 200 = 2 ¿PEG
á«fÉãdG Ö«°üf
QÉæjO 1000 = 200 * 5 = á«fÉãdG Ö«°üf É¡æeh 200 = 5
(ófQ + á°ûFÉY + óªMCG) Ö«°üf ófQ Ö«°üf á°ûFÉY Ö«°üf óªMCG Ö«°üf
25 = 625 =
25 5 + 8 + 12 = 5 = 8 = 12
(10
óªMCG Ö«°üf
QÉæjO 300 = 25 * 12 = óªMCG Ö«°üf É¡æeh 25 = 12
á°ûFÉY Ö«°üf
QÉæjO 200 = 25 * 8 = ºdÉ°S Ö«°üf É¡æeh 25 = 8
ófQ Ö«°üf
GQÉæjO 125 = 25 * 5 = ódÉN Ö«°üf É¡æeh 25 =
k 5
ôNBG πM
625 5+8+12
300 = 325 - 625 óªMC’ ≈≤Ñj É¡æeh 325 = ¢S ¢S = 5 + 8
325 5+8
125 ófQ Ö«°üf É¡æeh 125 = ¢U ¢U = 5
200 á°ûFÉY Ö«°üf ¿PEG
251
18. »JGP QÉÑàNG
(77 - 75) áëØ°U
áHÉLE’G áHÉLE’G õeQ Iô≤ØdG ºbQ áHÉLE’G áHÉLE’G õeQ Iô≤ØdG ºbQ (1
5 24
8 Ü 6 40 Ü 1
2
º°S4 1 Ü 7 5 , 3 , 3 , 1 CG 2
6 6 5 8 3
27 Ü 8 0^5 `L 3
8
º°S 7 øe ô¨°UG Ü 9 8- O 4
15 7 - CG 5
8
7- > 5 1 4 2 4- >1^45-> 1 1-
2 8 - > 5^3- ( `L 5 > 3 > 0^6 ( Ü 3 2 ( CG (2
1936 = 280 + 1656
70 70 = 4 + 23 * 72 = 4 + 2 + 7 * 3 * 7^2 = 4 + 2- - 3 * 7^2 ( CG (3
70 7 10 7 7
8- = 16- = 4- * 4 = 5 * 2 - 6 * 4 = 2- + 1 4 ( Ü
15 30 30 6*5 6 5
4 1 = 4 + 1 = 10 * 2 + 1 = (4^3 + 5^7) * 2 + 1 ( `L
5 5 5 5 5 5
10 21 = 1 5 + 9 1 = 1 5 + 2 * 4 3 = 1 5 + 0^5 _ 4 3 ( O
22 11 2 11 4 11 4
461 = 39 - 500 = 39 - 50 = 6 * 13 - 50 = 35 _ 13 - 50 = 5 5 _ 3 1 - 7 1
( `g
70 70 70 7 35 * 4 7 6 4 7 6 4 7
1
.Æ pQÉa Ap ÉfE’G ¿Rh óL ,≠c2^25 ¬Ø°üfp ≈dpEG Aƒ∏ªe ƒgh ¿ põjn h ,≠c 3 5 Ap ɪdÉH Aƒ∏ªe ƒgh AÉYh ¿ põj (4
m n rn r p pp r l n n o n l n l o
1
0^95 = 2^25 - 3^2 = 2^25 - 3 5 = πFÉ°ùdG ∞°üf ¿Rh
≠c 1^3 = 0^95 - 2^25 = ÆQÉa AÉYƒdG ¿Rh
Ω 150 = º°S 15000 = 5000 * 3 = ø«à°SQóªdG ø«H »≤«≤ëdG ó©ÑdG (5
68 : 8 ƒg º°SôdG ¢SÉ«≤e (6
68
º°S 51 = 17 * 3 = * 6 = ô«ÑµàdG ó©H IQƒ°üdG ¢VôY
8
¢S
»fÉãdG íHQ QÉæjO 500 = 6000 = ¢S = 10000 (7
12 600 12000
¢S
∫hC’G íHQ QÉæjO 400 = 4000 = ¢S = 8000
10 500 10000
252
19. ∂JÉeƒ∏©e ÈàNG äÉHÉLEG øjQɪàdGh á∏Ä°SC’G πM
(81 - 80) äÉëØ°üdG á«fÉãdG IóMƒ∏d
(1
áHÉLE’G áHÉLE’G õeQ Iô≤ØdG ºbQ áHÉLE’G áHÉLE’G õeQ Iô≤ØdG ºbQ
3252^4 CG 7 4502 Ü 1
7^854 CG 8 49 O 2
25 CG 9 3,2 O 3
9
8 Ü 10 32- O 4
2 O 11 CG * Ü = Ü * CG Ü 5
5 * (2 * 3) `L 6
7 * 32 = 56 , 52 = 32 , 13 * 5 = 65 , 25 * 32 = 200 (2
5 * 2(3-) = 5 * 5 * 5 * 3- * 3- ( CG (3
3
(2-) = 8- ( Ü
3
3
(¢S-) = ¢S- * ¢S- * ¢S- (`L
253
20. (1 - 2) πFÉ°ùeh øjQÉ“
(88)áëØ°U
6´ = 3 × 2´ = 3( 2´) 5¢U = 4 + 1¢U = 4¢U × ¢U
( Ü ( CG (1
4 4
5 5 6
410 = 10 ( O Ü × 3CG = 3(2Ü CG) ( `L
(Ü ( CG (2
(O ( `M
(h ( `g
(R
`g+¿
(0^3) ( Ü 5
¢S ( CG (3
6
(0^4) ( O ¢U 4¢S 12 ( `L
3
6 6
9
Ü CG 8- ( h CG ( `g
6
3
6
4 (R
12
¢U 6¢S 27 = 3(4¢U 2¢S 3) = ∫hC’G Ö©µªdG ºéM (4
¢U 3¢S 0^125 = 3(¢U ¢S 0^5) = »fÉãdG Ö©µªdG ºéM
3
6
2 = 3+32 = 32 * 32 = 2(32) ( CG (5
2 = 2+2+22 = 22 * 22 * 22 = 3(22)
6
( 2) = 2(32) ¿PEG
3 2
0^0036 = 2(0^06) = 2(0^3 * 0^2) ( Ü
0^0036 = 2(0^06) = 2(0^2 * 0^3)
(0^2 * 0^3) = 2(0^3 * 0^2) ¿PEG
2
254
21. (2 - 2) πFÉ°ùeh øjQÉ“
(94) áëØ°U
(Ü ( CG (1
( O ( `L
( h ( `g
( R
1
¿ (Ü ¢S ( CG (2
5
5
1 ( O Ω ( `L
3
8
`g
1 1
2¢U ( h 13 ( `g
¿
1
81 ( Ü 4 ( CG (3
9( O 38^4 ( `L
2000 ( `g
3
¢S 4 ( `L 7
¢U 3¢S ( Ü 5-
Ω 3 ( CG (4
4
10 * 9 + 310 * 7 + 210 * 8 + 110 * 5 + 010 * 4 = 97854 ( CG (5
10 * 1 + 510 * 0 + 410 * 0 + 310 * 0 + 210*0+ 110* 0+ 010*0=1000000 ( Ü
6
1
10 * 4 + 010 * 2 + 1-10* 3 + 2-10 * 5 = 42^35 ( `L
4
10 * 7- + 310 * 4- + 210 * 3- + 110* 5- + 010 * 6- = 74356- ( O
(10¢S , 5-¢S) , (9¢S , 4-¢S) , (8¢S , 3-¢S) , (7¢S , 2-¢S) , (6¢S , 1-¢S) , (4¢S , ¢S) (6
255
22. (3 - 2) πFÉ°ùeh øjQÉ“
(98) äÉëØ°üdG
3 = ¢S ⇐ 32 = 2 * 2 * 2 = ¢S2 ⇐ 8 = ¢S2 ( CG (1
5 = ¢U ⇐ 5(0^010) = ¢U(0^01) ( Ü
1 1 ∫ 1
1=∫ ⇐ = 0^5 = ( `L
2 2
0 = ¢S ⇐ 05 = ¢S5 ⇐ 1 = ¢S5 ( O
4
(0^1) = 0^1 * 0^1 * 0^1 * 0^1 = (0^0001 ) = Ω(0^1) ( `g
4=Ω ⇐
2- = ¢S ⇐ 1 = 1 = ¢S3 ( h
2
3 9
3- 3 ¢S 3 3 4 64 3 3
3- = ¢S ⇐ 4 = 4 ⇐ 3 = 27 = 4 ( R
6 = ¿ ⇐ 62 = ¿2 ⇐ 64 = ¿2 (2
2 = (¿) ºbQ ™HôªdG øY áÄaɵªdG ᪫b (3
(1-¿)
kÉ°Tôb 32 = 52 = ¢SOÉ°ùdG ™HôªdG »a ≠∏ѪdG ( CG
kÉ°Tôb 512 = 92 = ô°TÉ©dG ™HôªdG »a ≠∏ѪdG ( Ü
5
2 = 2 * 2 * 2 * 2 * 2 = 1-¢S2 ⇐ 32 = 1-¢S 2 ( `L
5 = 1 - ¢S ⇐
6 = ¢S ¿PEG
2 = ¿ƒà°ùdGh ådÉãdG ™HôªdG »a ≠∏ѪdG ( O
62
¢Thôb 4611686018427387904 =
= ≈dhC’G Iô°û©dG äÉ©HôªdG øe ¬«∏Y π°üM …òdG ≠∏ѪdG ´ƒªée (`g
kÉ°Tôb 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1
256
23. (4 - 2) πFÉ°ùeh øjQÉ“
(104) áëØ°U
3 * 3 * 7 * 7 * 5 * 5 = 11025 ( CG (1
105 = 3 * 7 * 5 = 11025 ¿PEG
11 * 11 * 2 * 2 * 2 * 2 * 5 * 5 = 48400 (Ü
220 = 11 * 2 * 2 * 5 = 48400 ¿PEG
9 3*3 3*3*3*3 81
= 5*5*2*2 = 5 * 5 * 5 * 5 * 2 * 2 * 2 * 2 = 10000 = 0^0081 ( `L
100
0^09 =
7 7 7*7 49
0^875 = 8 = 2 * 2 * 2 = 2 * 2 * 2 * 2 * 2 * 2 = 64 (O
5 5 5*5 25 9 ( `g
1^25 = 4 = 2 * 2 = 2 * 2 * 2 * 2 = 16 = 1 16
210 * 4 = 410 * 16 = 10 * 16
4
= 510 * 1^6 (h
100 > 85 > 81 ¿CG ɪH ( CG (2
100 > 85 > 81 ¿PEG
10 > 85 >9
, 9^1 ≈橪H 9^5 , 9 ™≤j 85 ¿CG …CG 9 Oó©∏d ÜôbCG ¬æµdh 10 , 9 øjOó©dG ø«H ô°üëæj 85 Oó©dG
¬d ájôjó≤J ɪ«b 9^4 , 9^3 , 9^2
k
144 > 130 > 121 ¿CG ɪH (Ü
144 > 130 > 121 ¿PEG
12 > 130 >11
, 11^3 , 11^2 , 11^1 ¿CG ≈橪H 11 Oó©∏d ÜôbCG ¬æµdh 12 , 11 øjOó©dG ø«H ô°üëæj 130 Oó©dG
.¬d ájôjó≤J ɪ«b 11^4
k
257
24. (4 - 2) πFÉ°ùeh øjQÉ“ ™HÉJ
(104)áëØ°U
1521 > 1500 > 1444 ¿CG ɪH ( `L
1521 > 1500 > 1444 ¿PEG
39 > 1500 >38
, 38^8 , 38^7 , 38^6 ¿CG …CG 39 Oó©∏d ÜôbCG ¬æµdh 39 , 38 øjOó©dG ø«H ô°üëæj 1500 Oó©dG
Oó©∏d ájôjó≤J º«b »g 39^9
31 31 31
10 = = 100 = 0^31 (O
100
36 > 31 > 25 ¿CG ɪH
36 > 31 > 25 ¿PEG
6 > 31 >5
Oó©∏d ájôjó≤J º«b 5^9 , 5^8 , 5^7 , 5^6 º«≤dG ¿PEG 6 Oó©∏d ÜôbCG ¬æµdh 6 , 5 ø«H ô°üëæj 31 Oó©dG
0^31 Oó©∏d ájôjó≤J º«b »g ,0^59 ,0^58 ,0^57 ,0^56 ¿CG óéf 10 ≈∏Y º«≤dG √òg ᪰ù≤Hh 31
60 60 60
= = 10000 =0^006 (`g
100 10000
64 > 60 > 49 ¿CG ɪHh
¿EÉa
64 > 60 > 49
8 > 60 > 7 ¿CG …CG
√òg øe πc ᪰ù≤Hh 60 Oó©∏d ájôjó≤J º«b »g 7^9 , 7^8 , 7^7 , 7^6 º«≤dG ¿EÉa 64 ≈dEG ÜôbCG 60 ¿CG ɪHh
0^006 Oó©∏d ájôjó≤J º«b »g 0^079 , 0^078 , 0^077 , 0^076 º«≤dG ≈∏Y π°üëf 100 ≈∏Y º«≤dG
2
10 * 5 = ™∏°†dG ∫ƒW ⇐ 2Ω2 10 * 5 = 2Ω 1000 * 0^5 = ºfhO 0^5 (3
2
10 * 5 =
10 * 5 =
2^4 , 2^3 , 2^2 , 2^1 :»g 5 Oó©∏d ájôjó≤àdG º«≤dG øµd
24 , 23 , 22 , 21 :»g ´QÉ°û∏d á¡LGƒªdG á¡édG ∫ƒ£d ájôjó≤àdG º«≤dG ¿PEG
Ω 5^5 = 1-10 * 55 = 2-
10 * 3025 = 30^25 (4
258
25. (5 - 2) πFÉ°ùeh øjQÉ“
(109) áëØ°U
15- = 15- * 15- * 15- 3
= 3(15-) 3
( CG (1
50 = 50 * 50 * 50 3
= 125000 3
(Ü
50 = 10 * 5 = 310 * 125 3
= 125000 3
hCG
4 64 3
64
0^4 = 10 = = = 0^064 3
( `L
1000 3 1000
3- 27- 3
1^5- = 2 = = 27- 3 =3 3 - 3 ( O
8 3 8 8
125 >96 > 64 ¿CG ɪH ( CG (2
q
125 3
> 96 3
> 64 3
¿PEG
ájôjó≤J º«b »g 4^9 , 4^8 , 4^7 , 4^6 ¿EÉa 64 ≈dEG ¬æe 125 ≈dEG ÜôbCG 96 ¿CG ɪHh 5 > 96
q q 3 >4
96 3
Oó©∏d
125 > 73 > 64 ¿CG ɪH ( Ü
q
125 3
> 73 3
> 64 3
¿PEG
5 > 73 3
>4
73 3 Oó©∏d ájôjó≤J º«b »g 4^4 , 4^3 , 4^2 , 4^1 ¿EÉa 125 ≈dEG ¬æe 64 ≈dEG ÜôbCG 73 ¿CG ɪHh
q q
64- > 86- > 125- ¿CG ɪH ( `L
q
64- 3
> 86- 3
> 125- 3
¿EÉa
4- > 86- 3
> 5-
ájôjó≤J º«b »g 4^4- , 4^3- , 4^2- , 4^1- OGóYC’G ¿PEG 125- ≈dEG ¬æe 64- ≈dEG ÜôbCG 86- øµd
86- 3
Oó©∏d
3
º°S 26 = ¢SÉëædG á©£b ºéM ¿C’ º©f ( CG (3
q
27 > 26 > 8 ¿CG ßMÓf ¬æeh
q
27 3
> 26
3
> 8
3
¿PEG
26 3
Oó©∏d ájôjó≤J º«b »g 2^9 , 2^8 , 2^7 , 2^6 OGóYC’G ¿EÉa 27 ≈dEG ÜôbCG 26 ¿CG ɪHh 3 > 26
q
3
>2
™æ°U øµªj ¬fCG èàæà°ùJ Gò¡Hh ,º°S 2^6 ¬aôM ∫ƒW ¿ƒµj ¢SÉëædG á©£b øe ¬©æ°U øµªj Ö©µe ô¨°UCG ¿CG …CG
q
.¢SÉëædG á©£b øe º°S 2^4 ¬aôM ∫ƒW Ö©µe
259
26. (5 - 2) πFÉ°ùeh øjQÉ“
(113) áëØ°U
7 6 ( CG ( 1
11 3
5 (Ü
3 8 = 3 4 + 3 2 - 3 6 = 16 * 3 + 4*3 - 36 * 3 (`L
( 11 3
- 3 3
- 3 )7(O
(3 3
+ 1) 14 ( CG ( 2
2 4 + 6 12 (Ü
12 3 12 8 12 3 12 3- 12 2 12
12
+
12
+
12
= ( 12 ) (
-
3
+
4 ) (`L
12 14
=
12
3*4 14
=
12
3 7
=
3
260
