Математика и информатика. Решение логико-познавательных задач. Учебное пособие для студентов вузов

ǺȈȉȓȐȞȈ 1.13

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ: 11200210112012113 462346549 .

1.5.ǨȘȐȜȔȍȚȐȟȍșȒȐȍ ȖȗȍȘȈȞȐȐ

ȊȗȖȏȐȞȐȖȕȕȣȝ șȐșȚȍȔȈȝ șȟȐșȓȍȕȐȧ

ȼɩɨɡɢɰɢɨɧɧɵɯ ɫɢɫɬɟɦɚɯ ɫɱɢɫɥɟɧɢɹ ɚɪɢɮɦɟɬɢɱɟɫɤɢɟ ɨɩɟɪɚɰɢɢ ɩɪɨɢɡɜɨɞɹɬɫɹ ɩɨ ɡɚɤɨɧɚɦ ɞɟɫɹɬɢɱɧɨɣ ɚɪɢɮɦɟɬɢɤɢ.

ɇɚɩɪɢɦɟɪ, ɜ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ, ɤɨɬɨɪɚɹ ɢɦɟɟɬ ɨɫɧɨɜɚɧɢɟ 2, ɡɚɩɢɫɶ ɱɢɫɟɥ ɩɪɨɢɡɜɨɞɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɞɜɭɯ ɰɢɮɪ (0, 1) ɜ ɨɬɥɢɱɢɟ ɨɬ ɞɟɫɹɬɢ ɰɢɮɪ ɞɟɫɹɬɢɱɧɨɣ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ.

Ɋɚɫɫɦɨɬɪɢɦ ɫɥɨɠɟɧɢɟ ɨɞɧɨɪɚɡɪɹɞɧɵɯ ɱɢɫɟɥ ɜ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɟ

ɫɱɢɫɥɟɧɢɹ. Ɋɚɜɟɧɫɬɜɚ 0 0 0; 0 1 1; 1 0 1 ɹɜɥɹɸɬɫɹ ɩɪɚɜɢɥɶ-

ɧɵɦɢ ɤɚɤ ɞɥɹ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɵ, ɬɚɤ ɢ ɞɥɹ ɞɟɫɹɬɢɱɧɨɣ ɫɢɫɬɟɦɵ. Ɉɞɧɚɤɨ ɪɟɡɭɥɶɬɚɬɵ ɫɥɨɠɟɧɢɹ 1 + 1 ɜ ɪɚɡɧɵɯ ɫɢɫɬɟɦɚɯ ɪɚɡɥɢɱɧɵ. ȼ ɞɟɫɹɬɢɱɧɨɣ ɫɢɫɬɟɦɟ ɪɟɡɭɥɶɬɚɬɨɦ ɹɜɥɹɟɬɫɹ 2, ɧɨ ɜ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɟ ɧɟɬ ɬɚɤɨɣ ɰɢɮɪɵ. ɂɡɜɟɫɬɧɨ, ɱɬɨ ɜ ɩɪɨɰɟɫɫɟ ɞɟɫɹɬɢɱɧɨɝɨ ɫɥɨɠɟɧɢɹ 9+1 ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɟɪɟɧɨɫ 1 ɜ ɫɬɚɪɲɢɣ ɪɚɡɪɹɞ, ɬɚɤ ɤɚɤ ɫɬɚɪɲɟ 9 ɰɢɮɪɵ ɧɟɬ, ɬ.ɟ. 9 1 10 . ȼ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɚɦɚɹ ɫɬɚɪɲɚɹ ɰɢɮɪɚ — 1, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, 1 1 10 . ɉɪɢ ɫɥɨɠɟɧɢɢ ɞɜɭɯ ɟɞɢɧɢɰ ɜ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɟ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɩɨɥɧɟɧɢɟ ɪɚɡɪɹɞɚ, ɩɨɷɬɨɦɭ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɟɪɟɧɨɫ ɜ ɫɬɚɪɲɢɣ ɪɚɡɪɹɞ. ɉɟɪɟɩɨɥɧɟɧɢɟ ɪɚɡɪɹɞɚ ɨɬɦɟɱɚɟɬɫɹ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɡɧɚɱɟɧɢɟ ɱɢɫɥɚ ɜ ɧɟɦ ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɵɦ ɢɥɢ ɛɨɥɶɲɢɦ ɨɫɧɨɜɚɧɢɹ. Ⱦɥɹ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɵ ɷɬɨ ɱɢɫɥɨ ɪɚɜɧɹɟɬɫɹ 2 (102 210 ).

ȼ ɩɪɨɰɟɫɫɟ ɫɥɨɠɟɧɢɹ ɦɧɨɝɨɪɚɡɪɹɞɧɵɯ ɱɢɫɟɥ ɞɟɣɫɬɜɭɸɬ ɬɟ ɠɟ ɩɪɚɜɢɥɚ, ɜ ɬɨɦ ɱɢɫɥɟ ɩɟɪɟɧɨɫ ɢɡ ɦɥɚɞɲɢɯ ɪɚɡɪɹɞɨɜ ɜ ɫɬɚɪɲɢɟ.

ɉɪɢɦɟɪ 26. ɇɚɣɞɟɦ ɫɭɦɦɭ ɱɢɫɟɥ 1012 ɢ 112 (ɪɢɫ. 1.8).

1012112

10002

ǸȐș. 1.8

ȼɵɩɨɥɧɹɟɦ ɫɥɨɠɟɧɢɟ ɩɨ ɩɪɚɜɢɥɚɦ ɞɟɫɹɬɢɱɧɨɣ ɚɪɢɮɦɟɬɢɤɢ. Ɋɚɫɫɭɠɞɚɟɦ ɬɚɤ:

12 12 102 , «ɧɨɥɶ» ɩɢɲɟɦ, «ɨɞɢɧ» ɞɟɪɠɢɦ ɜ ɭɦɟ (ɩɟɪɟɧɨɫɢɦ ɜ ɞɪɭɝɨɣ ɪɚɡɪɹɞ);

21

02 12 12 , ɬɟɩɟɪɶ ɞɨɛɚɜɥɹɟɦ «ɨɞɢɧ» (ɤɨɬɨɪɵɣ ɞɟɪɠɚɥɢ ɜ ɭɦɟ) ɢ ɩɨɥɭɱɚɟɦ: 12 12 102 . ɋɧɨɜɚ «ɧɨɥɶ» ɩɢɲɟɦ, ɚ «ɨɞɢɧ» ɞɟɪɠɢɦ ɜ ɭɦɟ ɢ ɬ.ɞ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɨɢɡɨɲɥɚ «ɰɟɩɧɚɹ ɪɟɚɤɰɢɹ», ɤɨɝɞɚ ɩɟɪɟɧɨɫ ɟɞɢɧɢɰɵ ɜ ɨɞɢɧ ɪɚɡɪɹɞ ɜɵɡɵɜɚɟɬ ɩɟɪɟɧɨɫ ɜ ɫɥɟɞɭɸɳɢɣ ɪɚɡɪɹɞ.

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ: 1012 112 10002 .

ȼɵɱɢɬɚɧɢɟ ɦɧɨɝɨɪɚɡɪɹɞɧɵɯ ɞɜɨɢɱɧɵɯ ɱɢɫɟɥ ɩɪɨɢɡɜɨɞɢɬɫɹ ɫ ɭɱɟɬɨɦ ɜɨɡɦɨɠɧɵɯ ɡɚɣɦɨɜ ɢɡ ɫɬɚɪɲɢɯ ɪɚɡɪɹɞɨɜ.

ɉɪɢɦɟɪ 27. ɇɚɣɞɟɦ ɪɚɡɧɨɫɬɶ ɱɢɫɟɥ 1012 ɢ 112 . ɂɦɟɟɦ (ɪɢɫ. 1.9):

1012

112

102

ǸȐș. 1.9

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ: 1012 112 102 .

ɍɦɧɨɠɟɧɢɟ ɢ ɞɟɥɟɧɢɹ ɱɢɫɟɥ ɜ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɟ ɜɵɩɨɥɧɹɟɬɫɹ ɩɨ ɨɛɳɟɩɪɢɧɹɬɵɦ ɞɥɹ ɩɨɡɢɰɢɨɧɧɵɯ ɫɢɫɬɟɦ ɩɪɚɜɢɥɚɦ.

ɉɪɢɦɟɪ 28. ɇɚɣɞɟɦ ɩɪɨɢɡɜɟɞɟɧɢɟ ɱɢɫɟɥ 1012 ɢ 112 (ɪɢɫ. 1.10).

υ1012 112

1012 1012

11112

ǸȐș. 1.10

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ: 1012 112 11112 .

Ⱥɪɢɮɦɟɬɢɱɟɫɤɢɟ ɞɟɣɫɬɜɢɹ ɜ ɞɪɭɝɢɯ ɩɨɡɢɰɢɨɧɧɵɯ ɫɢɫɬɟɦɚɯ ɫɱɢɫɥɟɧɢɹ ɜɵɩɨɥɧɹɸɬɫɹ ɚɧɚɥɨɝɢɱɧɨ. ɇɟɨɛɯɨɞɢɦɨ ɬɨɥɶɤɨ ɩɨɦɧɢɬɶ, ɱɬɨ ɩɟɪɟɧɨɫ ɜ ɫɥɟɞɭɸɳɢɣ ɪɚɡɪɹɞ ɩɪɢ ɫɥɨɠɟɧɢɢ ɢ ɡɚɟɦ ɢɡ ɫɬɚɪɲɟɝɨ ɪɚɡɪɹɞɚ ɩɪɢ ɜɵɱɢɬɚɧɢɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɜɟɥɢɱɢɧɨɣ ɨɫɧɨɜɚɧɢɹ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ.

ɉɪɢɦɟɪ 29. Ʉɚɤɨɟ ɰɟɥɨɟ ɱɢɫɥɨ ɫɥɟɞɭɟɬ ɡɚ ɱɢɫɥɨɦ 5 ɜ ɲɟɫɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ? ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɞɟɫɹɬɢɱɧɨɣ ɫɢɫɬɟɦɟ 5 1 6 , ɚ ɜ ɲɟɫɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɞɥɹ ɡɚɩɢɫɢ ɱɢɫɟɥ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜɫɟɝɨ ɲɟɫɬɶ ɰɢɮɪ: 0, 1, 2, 3, 4, 5 ɢ ɰɢɮɪɵ 6 ɧɟɬ, ɩɨɷɬɨɦɭ ɩɪɨɢɫɯɨɞɢɬ ɩɟ-

22

ɉɪɢɦɟɪ 30. Ʉɚɤɨɟ ɰɟɥɨɟ ɱɢɫɥɨ ɫɥɟɞɭɟɬ ɡɚ ɱɢɫɥɨɦ F ɜ ɲɟɫɬɧɚɞɰɚɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ? ȼ ɲɟɫɬɧɚɞɰɚɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɞɥɹ ɡɚɩɢɫɢ ɱɢɫɟɥ ɢɫɩɨɥɶɡɭɟɬɫɹ ɲɟɫɬɧɚɞɰɚɬɶ ɰɢɮɪ: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. ɉɨɷɬɨɦɭ, ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɢɫɥɚ F ɧɚ ɟɞɢɧɢɰɭ, ɩɪɨɢɫɯɨɞɢɬ «ɫɤɚ-

ɩɢɫɢ ɱɢɫɟɥ ɜ ɩɹɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɢɫɩɨɥɶɡɭɸɬɫɹ ɰɢɮɪɵ 0, 1, 2, 3, 4, ɬɨ ɩɟɪɟɩɨɥɧɟɧɢɟ ɪɚɡɪɹɞɚ ɩɪɨɢɫɯɨɞɢɬ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɢɫɥɚ 4 ɧɚ ɟɞɢɧɢɰɭ: 45 15 105 . ȼ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɩɪɢ ɭɦɟɧɶɲɟɧɢɢ ɱɢɫɥɚ 105 ɧɚ ɟɞɢɧɢɰɭ ɩɪɨɢɫɯɨɞɢɬ «ɜɨɡɜɪɚɳɟɧɢɟ» ɜ ɦɥɚɞɲɢɣ ɪɚɡɪɹɞ.

ɉɪɢɦɟɪ 32. Ʉɚɤɨɟ ɰɟɥɨɟ ɱɢɫɥɨ ɩɪɟɞɲɟɫɬɜɭɟɬ ɱɢɫɥɭ 20 ɜ ɜɨɫɶɦɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ (ɪɢɫ. 1.11)?

208 –18

178

ǸȐș. 1.11

Ɉɱɟɜɢɞɧɨ, ɱɬɨ 208 18 108 (108 18 ) 108 78 178 . ȼ ɜɨɫɶɦɟɪɢɱ-

ɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ ɫɬɚɪɲɟɣ ɰɢɮɪɨɣ ɹɜɥɹɟɬɫɹ ɰɢɮɪɚ 7, ɩɨɷɬɨɦɭ ɩɟɪɟɩɨɥɧɟɧɢɟ ɪɚɡɪɹɞɚ ɩɪɨɢɫɯɨɞɢɬ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɢɫɥɚ 78 ɧɚ ɟɞɢɧɢɰɭ, ɚ ɜɨɡɜɪɚɳɟɧɢɟɜɦɥɚɞɲɢɣɪɚɡɪɹɞɩɪɢɭɦɟɧɶɲɟɧɢɢ ɱɢɫɥɚ108 ɧɚɟɞɢɧɢɰɭ.

ȼ ɨɫɧɨɜɟ ɩɪɚɜɢɥ ɚɪɢɮɦɟɬɢɤɢ ɥɸɛɨɣ ɩɨɡɢɰɢɨɧɧɨɣ ɫɢɫɬɟɦɵ ɥɟɠɚɬ ɬɚɛɥɢɰɵ ɫɥɨɠɟɧɢɹ ɢ ɭɦɧɨɠɟɧɢɹ ɨɞɧɨɪɚɡɪɹɞɧɵɯ ɱɢɫɟɥ, ɚɧɚɥɨɝɢɱɧɵɟ ɬɚɛɥɢɰɚɦ ɚɪɢɮɦɟɬɢɱɟɫɤɢɯ ɨɩɟɪɚɰɢɣ ɜ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ (ɫɦ. ɩ. 1.3). Ɍɚɤɢɦɢ ɬɚɛɥɢɰɚɦɢ ɭɞɨɛɧɨ ɩɨɥɶɡɨɜɚɬɶɫɹ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɞɟɣɫɬɜɢɣ ɧɚɞ ɦɧɨɝɨɪɚɡɪɹɞɧɵɦɢ ɱɢɫɥɚɦɢ.

ɉɪɢɦɟɪ 33. ȼɵɱɢɫɥɢɦ 3425 235 ɜ ɩɹɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ. ɑɬɨɛɵ ɭɩɪɨɫɬɢɬɶ ɜɵɱɢɫɥɟɧɢɹ ɫɨɫɬɚɜɢɦ ɬɚɛɥɢɰɭ ɫɥɨɠɟɧɢɹ ɞɥɹ ɩɹɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɵ (ɬɚɛɥ. 1.14).

23

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ: 3425 235 4205 .

ɉɪɢɦɟɪ 34. ȼɵɱɢɫɥɢɦ 2135 35 ɜ ɩɹɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ. ɑɬɨɛɵ ɭɩɪɨɫɬɢɬɶ ɜɵɱɢɫɥɟɧɢɹ ɫɨɫɬɚɜɢɦ ɬɚɛɥɢɰɭ ɭɦɧɨɠɟɧɢɹ ɞɥɹ

ɩɹɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɵ (ɬɚɛɥ. 1.15).

ǺȈȉȓȐȞȈ 1.15

ɂɫɩɨɥɶɡɭɹ ɬɚɛɥɢɰɭ, ɜɵɱɢɫɥɹɟɦ 2135 35 (ɪɢɫ. 1.13).

24

υ 2135 35

11445

ǸȐș. 1.13

ɉɪɢɦɟɪ 35. ȼɵɱɢɫɥɢɦ: 178 1716 , ɨɬɜɟɬ ɡɚɩɢɫɚɬɶ ɜ ɜɨɫɶɦɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ.

ɉɪɢɜɟɞɟɦ ɱɢɫɥɨ 1716 ɤ ɨɫɧɨɜɚɧɢɸ 8 ɩɨɫɪɟɞɫɬɜɨɦ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ: 1716 0001 01112 010 1112 278 (ɩɪɨɛɟɥɚɦɢ ɭɫɥɨɜɧɨ

ɨɛɨɡɧɚɱɢɦ ɞɟɥɟɧɢɟ ɧɚ ɬɟɬɪɚɞɵ ɢ ɬɪɢɚɞɵ).

ȼɵɩɨɥɧɢɦ ɫɥɨɠɟɧɢɟ ɜ ɜɨɫɶɦɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ (ɪɢɫ.1.14).

178

278

468

ǸȐș. 1.14

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ: 178 1716 178 278 468 .

ɉɪɢɦɟɪ 36. ȼɵɱɢɫɥɢɦ 29F716 1316 ɜ ɲɟɫɬɧɚɞɰɚɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ.

ȼɵɩɨɥɧɹɟɦ ɫɥɨɠɟɧɢɟ ɫɬɨɥɛɢɤɨɦ (ɪɢɫ. 1.15).

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ: 29F516 1516 2A0A16 .

ɉɪɢɦɟɪ 37. Ɂɚɩɢɲɟɦ ɞɟɫɹɬɢɱɧɨɟ ɡɧɚɱɟɧɢɟ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɱɟɬɵɪɟɯɪɚɡɪɹɞɧɨɝɨ ɬɪɨɢɱɧɨɝɨ ɱɢɫɥɚ.

Ɉɱɟɜɢɞɧɨ, ɱɬɨ 22223 ɹɜɥɹɟɬɫɹ ɦɚɤɫɢɦɚɥɶɧɵɦ ɱɟɬɵɪɟɯɪɚɡɪɹɞɧɵɦ ɬɪɨɢɱɧɵɦ ɱɢɫɥɨɦ. ɉɟɪɟɜɟɞɟɦ ɞɚɧɧɨɟ ɱɢɫɥɨ ɜ ɞɟɫɹɬɢɱɧɭɸ ɫɢɫɬɟɦɭ ɫɱɢɫɥɟɧɢɹ:

25

22223 2 33 2 32 2 31 2 30 54 18 6 2 80.

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ: 22223 8010 .

ɉɪɢɦɟɪ 38. Ʉɚɤɨɟ ɧɚɢɛɨɥɶɲɟɟ ɞɟɫɹɬɢɱɧɨɟ ɱɢɫɥɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɪɟɦɹ ɰɢɮɪɚɦɢ ɜ ɲɟɫɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ?

Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɦɚɤɫɢɦɚɥɶɧɨɦɭ ɞɟɫɹɬɢɱɧɨɦɭ ɱɢɫɥɭ, ɤɨɬɨɪɨɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɪɟɦɹ ɰɢɮɪɚɦɢ ɜ ɲɟɫɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɛɭɞɟɬ ɫɨɨɬɜɟɬɫɬɜɨ-

ɉɪɢɦɟɪ 41. ɇɚɣɞɟɦ ɨɫɧɨɜɚɧɢɹ ɫɢɫɬɟɦ ɫɱɢɫɥɟɧɢɹ q ɢ p, ɜ ɤɨɬɨɪɵɯ ɫɩɪɚɜɟɞɥɢɜɨ ɪɚɜɟɧɫɬɜɨ: 12 p ! 21q .

Ɂɚɩɢɲɟɦ ɤɚɠɞɨɟ ɱɢɫɥɨ ɞɚɧɧɨɝɨ ɧɟɪɚɜɟɧɫɬɜɚ ɜ ɪɚɡɜɟɪɧɭɬɨɦ ɜɢɞɟ: 1 p1 2 p0 ! 2 q1 1 q0 . ɉɨɫɥɟ ɭɩɪɨɳɟɧɢɹ ɢɦɟɟɦ ɫɥɟɞɭɸɳɟɟ ɜɵɪɚɠɟɧɢɟ: 1 p 2 1 ! 2 q 1 1 . Ɂɧɚɱɢɬ: p ! 2 q 1 , ɨɬɤɭɞɚ ɩɨɥɭɱɚɟɦ, ɧɚɩɪɢɦɟɪ: p 10, q 5 .

ɉɪɢɦɟɪ 42. ȼ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ ɫ ɧɟɤɨɬɨɪɵɦ ɨɫɧɨɜɚɧɢɟɦ q ɞɟɫɹɬɢɱɧɨɟ ɱɢɫɥɨ 18 ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ 30. ɍɤɚɠɟɦ ɷɬɨ ɨɫɧɨɜɚɧɢɟ.

26

(q = 5)?

ɉɟɪɟɜɟɞɟɦ ɩɟɪɜɨɟ ɢ ɩɨɫɥɟɞɧɟɟ ɱɢɫɥɨ ɢɡ ɡɚɞɚɧɧɨɝɨ ɞɢɚɩɚɡɨɧɚ (ɱɢɫɥɚ 10 ɢ 18) ɢɡ ɞɟɫɹɬɢɱɧɨɣ ɜ ɩɹɬɟɪɢɱɧɭɸ ɫɢɫɬɟɦɭ ɫɱɢɫɥɟɧɢɹ: 10 205 , 18 335 . Ɂɚɩɢɲɟɦ ɩɨ ɩɨɪɹɞɤɭ ɱɢɫɥɚ ɨɬ 205 ɞɨ 335 :

205 , 215 , 225 , 235 , 245 , 305 , 315 , 325 , 335 .

Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɰɢɮɪɚ 2 ɜɫɟɝɨ ɜɫɬɪɟɱɚɟɬɫɹ ɫɟɦɶ ɪɚɡ.

ɉɪɢɦɟɪ 44. ȼ ɭɱɟɛɧɨɦ ɰɟɧɬɪɟ ɢɦɟɸɬɫɹ 214 ɩɟɪɫɨɧɚɥɶɧɵɯ ɤɨɦɩɶɸɬɟɪɨɜ — 120 ɩɟɪɜɨɝɨ ɬɢɩɚ ɢ 44 ɜɬɨɪɨɝɨ ɬɢɩɚ. ȼ ɤɚɤɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ ɩɨɫɱɢɬɚɧɵ ɤɨɦɩɶɸɬɟɪɵ?

ɂɦɟɟɦ: 214q 120q 44q , ɝɞɟ q — ɨɫɧɨɜɚɧɢɟ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ.

Ɂɚɩɢɲɟɦ ɤɚɠɞɨɟ ɱɢɫɥɨ ɞɚɧɧɨɝɨ ɪɚɜɟɧɫɬɜɚ ɜ ɪɚɡɜɟɪɧɭɬɨɦ ɜɢɞɟ:

2 q2 1 q1 4 q0 1 q2 2 q1 0 q0 4 q1 4 q0 .

ɉɨɫɥɟ ɭɩɪɨɳɟɧɢɹ ɩɨɥɭɱɚɟɦ ɤɜɚɞɪɚɬɧɨɟ ɭɪɚɜɧɟɧɢɟ: q2 5 q 0 . Ɋɟɲɚɹ ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɧɚɯɨɞɢɦ q 5 . Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɤɨɦɩɶɸɬɟɪɵ ɩɨɫɱɢɬɚɧɵ ɜ ɩɹɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ.

ɉɪɢɦɟɪ 45. ȼɫɟ ɩɹɬɢɛɭɤɜɟɧɧɵɟ ɫɥɨɜɚ, ɫɨɫɬɚɜɥɟɧɧɵɟ ɢɡ ɛɭɤɜ Ⱥ, ȼ, ɋ, ɍ, ɡɚɩɢɫɚɧɵ ɜ ɚɥɮɚɜɢɬɧɨɦ ɩɨɪɹɞɤɟ. ȼɨɬ ɧɚɱɚɥɨ ɫɩɢɫɤɚ:

1.ȺȺȺȺȺ

2.ȺȺȺȺȼ

3.ȺȺȺȺɋ

4.ȺȺȺȺɍ

5.ȺȺȺȼȺ

…………...

Ɂɚɩɢɲɟɦ ɫɥɨɜɨ, ɤɨɬɨɪɨɟ ɫɬɨɢɬ ɧɚ 250-ɦ ɦɟɫɬɟ ɨɬ ɧɚɱɚɥɚ ɫɩɢɫɤɚ. Ɂɚɦɟɧɢɦ ɛɭɤɜɵ Ⱥ, ȼ, ɋ, ɍ ɧɚ 0, 1, 2, 3 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɂɚɩɢɲɟɦ

ɧɚɱɚɥɨ ɫɩɢɫɤɚ, ɡɚɦɟɧɢɜ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɛɭɤɜɵ ɰɢɮɪɚɦɢ:

27

1.00000

2.00001

3.00002

4.00003

5.00010

…………...

Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ ɡɚɩɢɫɶ ɟɫɬɶ ɱɢɫɥɚ, ɡɚɩɢɫɚɧɧɵɟ ɜ ɩɨɪɹɞ-

ɤɟ ɜɨɡɪɚɫɬɚɧɢɹ ɜ ɱɟɬɜɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ. Ɍɨɝɞɚ ɧɚ 250 ɦɟɫɬɟ ɛɭɞɟɬ ɫɬɨɹɬɶ ɱɢɫɥɨ 249 (ɬɚɤ ɤɚɤ ɫɩɢɫɨɤ ɧɚɱɢɧɚɟɬɫɹ ɫ ɱɢɫɥɚ 00000),

ɡɚɩɢɫɚɧɧɨɟ ɜ ɱɟɬɜɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ. ɉɟɪɟɜɟɞɟɦ ɱɢɫɥɨ 249 ɜ ɱɟɬɜɟɪɢɱɧɭɸ ɫɢɫɬɟɦɭ ɫɱɢɫɥɟɧɢɹ (ɪɢɫ. 1.16).

249 : 4 62 1 (ɨɫɬɚɬɨɤ)

62 : 4 15 2 (ɨɫɬɚɬɨɤ)

15 : 4 3 3 (ɨɫɬɚɬɨɤ)

ǸȐș. 1.16

Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ: 24910 33214. Ɉɱɟɜɢɞɧɨ, ɱɬɨ 33214 — ɱɟɬɵɪɟɯɪɚɡɪɹɞɧɨɟ ɱɢɫɥɨ, ɚ ɩɨ ɭɫɥɨɜɢɸ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɩɹɬɢɪɚɡɪɹɞɧɨɟ ɱɢɫɥɨ (ɬɚɤ ɤɚɤ ɪɟɱɶ ɢɞɟɬ ɨ ɩɹɬɢɛɭɤɜɟɧɧɵɯ ɫɥɨɜɚɯ), ɩɨɷɬɨɦɭ ɩɪɢɩɢɲɟɦ ɤ ɞɚɧɧɨɦɭ ɱɢɫɥɭ ɫɥɟɜɚ 0 ɢ ɩɪɟɞɫɬɚɜɢɦ ɟɝɨ ɜ ɜɢɞɟ: 033214 . ɉɪɨɜɨɞɹ ɨɛɪɚɬɧɭɸ ɡɚɦɟɧɭ ɜ ɱɢɫɥɟ 033214 (ɰɢɮɪɵ ɡɚɦɟɧɹɟɦ ɧɚ ɛɭɤɜɵ), ɩɨɥɭɱɚɟɦ: Ⱥɍɍɋȼ.

ɉɪɢɦɟɪ 46. Ⱦɥɹ ɤɨɞɢɪɨɜɚɧɢɹ ɛɭɤɜ ɏ, Ʌ, Ɉ, Ⱦ ɪɟɲɢɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɜɨɢɱɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɱɢɫɟɥ 0, 1, 2 ɢ 3 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ (ɫ ɫɨɯɪɚɧɟɧɢɟɦ ɨɞɧɨɝɨ ɧɟɡɧɚɱɚɳɟɝɨ ɧɭɥɹ ɜ ɫɥɭɱɚɟ ɨɞɧɨɪɚɡɪɹɞɧɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ). Ʉɚɤɨɟ ɩɨɥɭɱɢɬɫɹ ɱɢɫɥɨ, ɟɫɥɢ ɬɚɤɢɦ ɫɩɨɫɨɛɨɦ ɡɚɤɨɞɢɪɨɜɚɬɶ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɛɭɤɜ ɏɈɅɈȾ ɢ ɪɟɡɭɥɶɬɚɬ ɡɚɩɢɫɚɬɶ ɜ ɲɟɫɬɧɚɞɰɚɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ?

ɉɪɟɞɫɬɚɜɢɦ ɞɚɧɧɵɟ ɜ ɭɫɥɨɜɢɢ ɛɭɤɜɵ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɢɦ ɱɢɫɥɚ ɜ ɞɜɨɢɱɧɨɦ ɤɨɞɟ (ɬɚɛɥ. 1.16).

ǺȈȉȓȐȞȈ 1.16

Ɂɚɤɨɞɢɪɭɟɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɛɭɤɜ ɏɈɅɈȾ ɞɜɨɢɱɧɵɦ ɤɨɞɨɦ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ ɱɢɫɥɨ 00100110112 . Ɍɟɩɟɪɶ ɩɪɢɜɟɞɟɦ ɱɢɫɥɨ 00100110112 ɤ ɨɫɧɨɜɚɧɢɸ 16, ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɨɬɤɢɧɭɜ ɫɥɟɜɚ ɞɜɚ ɧɟɡɧɚɱɚɳɢɯ ɧɭɥɹ: 100110112 9ȼ16 (ɩɪɨɛɟɥɚɦɢ ɭɫɥɨɜɧɨ ɨɛɨɡɧɚɱɢɦ ɞɟɥɟɧɢɟ ɧɚ ɬɟɬɪɚɞɵ).

DzȖȕȚȘȖȓȤȕȣȍ ȏȈȌȈȕȐȧ Ȓ ȋȓȈȊȍ 1

1.Ʉɚɤɨɟ ɦɢɧɢɦɚɥɶɧɨɟ ɨɫɧɨɜɚɧɢɟ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ ɦɨɠɟɬ ɢɦɟɬɶ ɱɢɫɥɨ B88 ?

2.ɇɚɣɞɢɬɟ ɦɢɧɢɦɚɥɶɧɨɟ ɨɫɧɨɜɚɧɢɟ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ, ɟɫɥɢ ɜ ɧɟɣ ɡɚɩɢɫɚɧɵ ɜɫɟ ɫɥɟɞɭɸɳɢɟ ɱɢɫɥɚ: 155, 146, 4401, 1010001, 335.

3.Ɂɚɩɢɲɢɬɟ ɞɟɫɹɬɢɱɧɨɟ ɡɧɚɱɟɧɢɟ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɲɟɫɬɢɪɚɡɪɹɞɧɨɝɨ ɬɪɨɢɱɧɨɝɨ ɱɢɫɥɚ.

4.Ʉɚɤɨɟ ɧɚɢɛɨɥɶɲɟɟ ɞɟɫɹɬɢɱɧɨɟ ɱɢɫɥɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɪɟɦɹ ɰɢɮɪɚɦɢ ɜ ɩɹɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ?

5.ȼɨ ɫɤɨɥɶɤɨ ɪɚɡ ɭɜɟɥɢɱɢɬɫɹ ɱɢɫɥɨ 3256 , ɟɫɥɢ ɩɪɢɩɢɫɚɬɶ ɤ ɧɟɦɭ ɫɩɪɚɜɚ ɨɞɢɧ ɧɨɥɶ?

6.Ʉɚɤɢɟ ɰɟɥɵɟ ɱɢɫɥɚ ɫɥɟɞɭɸɬ ɡɚ ɱɢɫɥɚɦɢ 1012 , 1778 , 9FF16 ?

7.Ʉɚɤɢɟ ɰɟɥɵɟ ɱɢɫɥɚ ɩɪɟɞɲɟɫɬɜɭɸɬ ɱɢɫɥɚɦ 10102 , 308 , 6C016 ?

8.ɉɟɪɟɜɟɞɢɬɟɱɢɫɥɨ 10101001102 ɜɱɟɬɜɟɪɢɱɧɭɸɫɢɫɬɟɦɭɫɱɢɫɥɟɧɢɹ.

9.ɉɟɪɟɜɟɞɢɬɟ ɱɢɫɥɨ 101204131422125 ɜ 25-ɪɢɱɧɭɸ ɫɢɫɬɟɦɭ ɫɱɢɫɥɟɧɢɹ.

10.ȼɵɱɢɫɥɢɬɟ 2456 3116 ɜ ɲɟɫɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ.

11.ȼɵɱɢɫɥɢɬɟ 10007 5117 ɜ ɫɟɦɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ.

12.ɇɚɣɞɢɬɟ ɨɫɧɨɜɚɧɢɟ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ q, ɜ ɤɨɬɨɪɨɣ ɫɩɪɚɜɟɞɥɢɜɨ

ɪɚɜɟɧɫɬɜɨ: 21q 24q 100q .

13.ɇɚɣɞɢɬɟ ɨɫɧɨɜɚɧɢɟ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ q, ɜ ɤɨɬɨɪɨɣ ɫɩɪɚɜɟɞɥɢɜɨ ɪɚɜɟɧɫɬɜɨ: 53q 14q 100q .

14.ȼ ɫɚɞɭ 100 ɮɪɭɤɬɨɜɵɯ ɞɟɪɟɜɶɟɜ — 14 ɹɛɥɨɧɶ ɢ 42 ɝɪɭɲɢ. ȼ ɤɚɤɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ ɩɨɫɱɢɬɚɧɵ ɞɟɪɟɜɶɹ?

15.ȼ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ ɫ ɧɟɤɨɬɨɪɵɦ ɨɫɧɨɜɚɧɢɟɦ q ɞɟɫɹɬɢɱɧɨɟ ɱɢɫɥɨ 162 ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ 200. ɍɤɚɠɢɬɟ ɷɬɨ ɨɫɧɨɜɚɧɢɟ.

16.ȼɫɟ ɩɹɬɢɛɭɤɜɟɧɧɵɟ ɫɥɨɜɚ, ɫɨɫɬɚɜɥɟɧɧɵɟ ɢɡ ɛɭɤɜ Ⱥ, Ɉ, ɍ, ɡɚɩɢɫɚɧɵ ɜ ɚɥɮɚɜɢɬɧɨɦ ɩɨɪɹɞɤɟ. ȼɨɬ ɧɚɱɚɥɨ ɫɩɢɫɤɚ:

1.ȺȺȺȺȺ

2.ȺȺȺȺɈ

3.ȺȺȺȺɍ

4.ȺȺȺɈȺ

…………...

29

Ɂɚɩɢɲɢɬɟ ɫɥɨɜɨ, ɤɨɬɨɪɨɟ ɫɬɨɢɬ ɧɚ 210-ɦ ɦɟɫɬɟ ɨɬ ɧɚɱɚɥɚ ɫɩɢɫɤɚ.

17.Ⱦɥɹ ɤɨɞɢɪɨɜɚɧɢɹ ɛɭɤɜ Ɉ, ȼ, Ⱦ, ɉ, Ⱥ ɪɟɲɢɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɜɨɢɱɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɱɢɫɟɥ 0, 1, 2, 3 ɢ 4 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ (ɫ ɫɨɯɪɚɧɟɧɢɟɦ ɨɞɧɨɝɨ ɧɟɡɧɚɱɚɳɟɝɨ ɧɭɥɹ ɜ ɫɥɭɱɚɟ ɨɞɧɨɪɚɡɪɹɞɧɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ). Ʉɚɤɨɟ ɩɨɥɭɱɢɬɫɹ ɱɢɫɥɨ, ɟɫɥɢ ɬɚɤɢɦ ɫɩɨɫɨɛɨɦ ɡɚɤɨɞɢ-

ɪɨɜɚɬɶ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɛɭɤɜ ȼɈȾɈɉȺȾ ɢ ɪɟɡɭɥɶɬɚɬ ɡɚɩɢɫɚɬɶ ɜ ɜɨɫɶɦɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ?

18.ɋɤɨɥɶɤɨ ɜɫɟɝɨ ɪɚɡ ɜɫɬɪɟɱɚɟɬɫɹ ɰɢɮɪɚ 4 ɜ ɡɚɩɢɫɢ ɱɢɫɟɥ 20, 21, 22, 23, 24, 25, 26, 27, ɟɫɥɢ ɷɬɢ ɱɢɫɥɚ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɲɟɫɬɟɪɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ?

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