Question

Сколько чётных шестизначных чисел, делящихся на 15, сумма цифр которых не более 4?

Answer 1

6/Задание № 1:

Сколько чётных шестизначных чисел, делящихся на 15, сумма цифр которых не более 4?

РЕШЕНИЕ: Так как число четное, то оно делится на 2. Кроме этого, так как число делится на 15, то оно делится на 3 и на 5. То есть число оканчивается нулем, и сумма его цифр делится на 3.

Очевидно, что сумма цифр не может равняться нулю. Кроме этого, если сумма цифр не более 4, то единственный допустимый вариант того, чтобы она делилась на 3 - это сумма 3.

Варианты: 300000, 210000, 201000, 200100, 200010, 120000, 102000, 100200, 100020, 111000, 110100, 110010, 101100, 101010, 100110.

ОТВЕТ: 15 чисел

Answer 2

Calculation of Even Six-Digit Numbers Divisible by 15 with a Digit Sum Not Exceeding 4

To calculate the number of even six-digit numbers that are divisible by 15 and have a digit sum not exceeding 4, we can break down the problem into smaller steps.

Step 1: Divisibility by 15 To be divisible by 15, a number must be divisible by both 3 and 5. Since we are looking for even numbers, the last digit must be either 0, 2, 4, 6, or 8. Therefore, the last digit of the six-digit number must be 0 or 5.

Step 2: Digit Sum To find the digit sum of a number, we need to add up all its digits. Since we want the digit sum to not exceed 4, the possible combinations are limited. Let's consider the cases:

- If the last digit is 0, the remaining five digits must have a sum of 4 or less. - If the last digit is 5, the remaining five digits must have a sum of 3 or less.

Step 3: Calculation Let's calculate the number of possibilities for each case:

Case 1: Last digit is 0 - The remaining five digits must have a sum of 4 or less. - We can use the concept of stars and bars to calculate the number of combinations. In this case, we have 5 stars (representing the five digits) and 4 bars (representing the sum limit of 4). - Using the stars and bars formula, the number of combinations is given by (5+4-1) choose (4) = 8 choose 4 = 70.

Case 2: Last digit is 5 - The remaining five digits must have a sum of 3 or less. - Using the stars and bars formula, the number of combinations is given by (5+3-1) choose (3) = 7 choose 3 = 35.

Step 4: Total Number of Possibilities To find the total number of possibilities, we add the number of possibilities from each case: - Total number of possibilities = 70 (from Case 1) + 35 (from Case 2) = 105.

Therefore, there are 105 even six-digit numbers that are divisible by 15 and have a digit sum not exceeding 4.