Предмет: Математика, автор: balikyanmaria

Найдите наименьшее натуральное число, кратное 19 и дающее остаток 1 при делении на каждое из чисел 3, 4 и 5.

Ответы

Автор ответа: GoldenVoice
2

Ответ:

361

Пошаговое объяснение:

Числа 3, 4 и 5 взаимно простые. Число 3\cdot4\cdot5=60 и, соответственно, число вида 60k, k\in\rm{Z}, делится на каждое из них без остатка, тогда число вида 60k+1 будет давать при делении на 3, 4, 5 остаток 1.

Таким образом, задача сводится к решению в целых числах уравнения 60k+1=19n либо 19n-60k=1.

Решим это уравнение.

n = \displaystyle\frac{{60k + 1}}{{19}} = 3k + \displaystyle\frac{{3k + 1}}{{19}}.

Обозначим

{t_1} = \displaystyle\frac{{3k + 1}}{{19}},

тогда

19{t_1} = 3k + 1,

3k = 19{t_1} - 1,

k = \displaystyle\frac{{19{t_1} - 1}}{3} = 6{t_1} + \displaystyle\frac{{{t_1} - 1}}{3}.

Обозначим

{t_2} = \displaystyle\frac{{{t_1} - 1}}{3},

тогда

3{t_2} = {t_1} - 1,

{t_1} = 3{t_2} + 1.

Подставляя найденное значение в выражение для k, получаем:

k = \displaystyle\frac{{19{t_1} - 1}}{3} = \displaystyle\frac{{19(3{t_2} + 1) - 1}}{3} = 19{t_2} + 6.

Тогда

n = \displaystyle\frac{{60k + 1}}{{19}} = \displaystyle\frac{{60(19{t_2} + 6) + 1}}{{19}} = 60{t_2} + 19.

Получили выражение для k и n, которые дают пары целых решений рассматриваемого уравнения при любых целых значениях {t_2}.

Очевидно, что при {t_2} = 0 значение n будет минимальным натуральным.

Следовательно, искомое число равно 19n = 19 \cdot 19 = 361.


antonovm: можно и так : число , уменьшенное на 1 кратно 60 , а увеличенное на 1 кратно 19 , далее перебор : 60 ; 120 ;180 ; 240 ; 300 ; 360
antonovm: нет , найти число , кратное 60 , кторое увеличенное на 1 кратно 19 , далее перебор
antonovm: моет быть так : 19n - 60k = 1 ; 19*19 -60*6 = 1 , вычитая из первого второе , получим : 19(n-19 ) = 60(k-6 ) => k-6 = 19x ; n-19 = 60x
antonovm: это чтобы не испоьзовать общую формулу для решения линейных диофантовых уравнений
Похожие вопросы
Предмет: Английский язык, автор: юля10293810
Предмет: Английский язык, автор: DII666


TEST 3. FORM 6th

Variant I

Task 1.Fill in the gaps with the proper words (Gifts, quiet, religious, hang).

1.Easter is a very important__________festival.

2.Where did you __________a clock? In the kitchen?

3.I had a lot of_______for Christmas.

4.Please, be ________. You are making so much noise!


Task.2 Replace the word that

1.Please, put the vase on the table that is standing by the bed.

2.Did you see the DVD disk that you had gave me for birthday?

3.I don’t like the teacher that gave me a bad mark.

with who or which

Task 2. Put sentences together to make one sentence.

1.Where does he go every Sunday? Tell me, please.

2.How did you do it? I’d like to know.

3.Do they play football well? We don’t know.

Task 3.Use the prepositions in the right place. (at, on, in)

1.The Great Fire of London was______ 1666.

2.I haven"t seen Tom for a few days. I saw him______Tuesday.

3.What do they do _____the evening.

4.If there are no clouds in the sky, you can see the stars ______night.

5.I always wear a warm clothes______a cold day.

Task 4. Read the text and match the names of the festival with the descriptions. There is

one extra.

A) St.Valentine’s Day

B) New Year

C) Christmas

D) Halloween

E) Easter

F) Mother's Day

1. On this day people visit their mothers and give them flowers and small presents.

Husbands and children help with the meals and washing up.

2. On this day people usually visit their friends. They try to be the first person to wish

good luck to their friends in the new year. There is a lot of dancing and eating. At

midnight everybody joins hands and sings a special song.

3. On this day people send cards to their friends, parents and relatives. Children get up

early in the morning. They want to see a stocking full of small presents on their beds
4. People send special cards to people they love. They shouldn't write their names on the
cards. The person who gets the card must guess who sent it.
5. On this day they say ghosts and witches come out. Children make lanterns out of
pumpkins. Some people have parties and dress as witches and ghosts. It is celebrated on
the 31st of October. Children walk from house to house and ask "Trick or treat?

4. People send special cards to people they love. They shouldn't write their names on the
cards. The person who gets the card must guess who sent it.
5. On this day they say ghosts and witches come out. Children make lanterns out of
pumpkins. Some people have parties and dress as witches and ghosts. It is celebrated on
the 31st of October. Children walk from house to house and ask "Trick or treat?