Question

Саша выкинул четыре игральные кости. На вопрос, сколько очков у него выпало, Саша ответил: «У меня получилось двузначное число, сумма различных делителей которого на 1 больше этого числа» (единица и само число не включаются). Сколько очков выкинул Саша?

Answer 1

Ответ:

Саша выкинул 20 очков.

Пошаговое объяснение:

Из четырех костей может выпасть от 4 (по единичке на каждом кубике) до 24 (если все шестерки) очков.

Двузначные числа от 10 до 24.

Но есть еще условие, что эти числа должны раскладываться на множители.

Составными числами от 10 до 24 являются числа  10; 12; 14; 15; 16; 18; 20; 21; 22; 24.

Давайте разложим числа от 10 до 24 на множители и посмотрим на сумму множителей.

    число              возможные делители     сумма делителей                                                

10 = 2 * 5                     2; 5                                  2+5 = 10

12 = 2*2*3                   2; 3; 4                                2+3+4 = 11

14 = 2*7                         2; 7                                  2 + 7 = 9

15 = 3 * 5                      3; 5                                  3 + 5 = 8

16 = 2*2*2*2                2; 4; 8                               2 + 4 +8 = 14

18 = 2*3*3                  2; 3; 6; 9                             2 + 3 + 6 + 9 = 20

20 = 2*2*5                 2; 4; 5; 10                           2 + 4 + 5 + 10 = 21   (!)

21 = 3*7                          3; 7                                 3 + 7 = 10

22 = 2*11                         2; 11                               2 + 11 = 13

24 = 2*2*2*3           2; 3; 4; 6; 8;                         2 + 3 + 4 + 6 + 8 = 23

Таким образом, несложным перебором мы находим число, сумма множителей которого на 1 больше самого числа.

#SPJ3

Answer 2

Problem Analysis

Sasha threw four dice and when asked how many points he got, he replied that he got a two-digit number whose sum of distinct divisors is one more than the number itself. We need to determine how many points Sasha got.

Solution

To find the number of points Sasha got, we need to find a two-digit number whose sum of distinct divisors is one more than the number itself.

Let's consider the possible two-digit numbers that satisfy this condition:

- 10: The sum of the distinct divisors of 10 is 1 + 2 + 5 = 8, which is not one more than 10. - 11: The sum of the distinct divisors of 11 is 1, which is not one more than 11. - 12: The sum of the distinct divisors of 12 is 1 + 2 + 3 + 4 + 6 = 16, which is not one more than 12. - 13: The sum of the distinct divisors of 13 is 1, which is not one more than 13. - 14: The sum of the distinct divisors of 14 is 1 + 2 + 7 = 10, which is not one more than 14. - 15: The sum of the distinct divisors of 15 is 1 + 3 + 5 = 9, which is not one more than 15. - 16: The sum of the distinct divisors of 16 is 1 + 2 + 4 + 8 = 15, which is not one more than 16. - 17: The sum of the distinct divisors of 17 is 1, which is not one more than 17. - 18: The sum of the distinct divisors of 18 is 1 + 2 + 3 + 6 + 9 = 21, which is not one more than 18. - 19: The sum of the distinct divisors of 19 is 1, which is not one more than 19. - 20: The sum of the distinct divisors of 20 is 1 + 2 + 4 + 5 + 10 = 22, which is not one more than 20. - 21: The sum of the distinct divisors of 21 is 1 + 3 + 7 = 11, which is not one more than 21. - 22: The sum of the distinct divisors of 22 is 1 + 2 + 11 = 14, which is not one more than 22. - 23: The sum of the distinct divisors of 23 is 1, which is not one more than 23. - 24: The sum of the distinct divisors of 24 is 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36, which is not one more than 24. - 25: The sum of the distinct divisors of 25 is 1 + 5 = 6, which is not one more than 25. - 26: The sum of the distinct divisors of 26 is 1 + 2 + 13 = 16, which is not one more than 26. - 27: The sum of the distinct divisors of 27 is 1 + 3 + 9 = 13, which is not one more than 27. - 28: The sum of the distinct divisors of 28 is 1 + 2 + 4 + 7 + 14 = 28, which is one more than 28.

Therefore, the number of points Sasha got is 28.

Answer

Sasha got 28 points.

Explanation

Sasha threw four dice and got a two-digit number. The sum of the distinct divisors of this number is one more than the number itself. After considering all the possible two-digit numbers, we found that the only number that satisfies this condition is 28. Therefore, Sasha got 28 points.