Question

Первая слева цифра шестизначного числа – 1. Если ее поставить на последнее место, то получится число, которое в 3 раза больше первоначального. Найти первоначальное число. ОТВЕТ С ПОЯСНЕНИЯМИ А НЕ ПРОСТО ЧИСЛО

Answer 1

Представим число как 1abcde, тогда:
300000+30000а+3000b+300c+30d+3e=100000a+10000b+1000c+100d+10e+1, отсюда 70000a+7000b+700c+70d+7e=299999, соответственно а можно взять только 4 раза, b - 2, c - 8, d - 5, e - 7. Ответ: 142857

Answer 2

Problem Analysis

We are given a six-digit number, where the first digit from the left is one less than the last digit. If we move the first digit to the last position, the resulting number is three times larger than the original number. We need to find the original number.

Solution

Let's assume the original number is ABCDEF, where A is the first digit and F is the last digit. According to the given information, we have the following equations:

1. A = F - 1 2. ABCDEF * 10 + A = 3 * ABCDEF

To solve this problem, we can substitute the value of A from equation 1 into equation 2 and solve for ABCDEF.

Detailed Solution

Let's substitute the value of A from equation 1 into equation 2:

(F - 1)BCDEF * 10 + (F - 1) = 3 * (F - 1)BCDEF

Simplifying the equation:

(F - 1)BCDEF * 10 + F - 1 = 3 * (F - 1)BCDEF

10(F - 1)BCDEF + F - 1 = 3(F - 1)BCDEF

10FBCDEF - 10BCDEF + F - 1 = 3FBCDEF - 3BCDEF

9FBCDEF - 10BCDEF + F - 1 = 0

Combining like terms:

9FBCDEF + F - 10BCDEF - 1 = 0

Rearranging the equation:

9FBCDEF + F = 10BCDEF + 1

Factoring out F and 10:

F(9BCDEF + 1) = 10BCDEF + 1

Dividing both sides by (9BCDEF + 1):

F = (10BCDEF + 1) / (9BCDEF + 1)

Since F is a digit, we can try different values for B, C, D, E, and F to find the solution.

Checking Different Values

Let's check different values for B, C, D, E, and F to find the solution:

1. Let's assume B = 0, C = 0, D = 0, E = 0, and F = 1:

F = (10 * 0 * 0 * 0 * 1 + 1) / (9 * 0 * 0 * 0 * 1 + 1) = (0 + 1) / (0 + 1) = 1 / 1 = 1

Since F = 1, this is a valid solution.

2. Let's assume B = 0, C = 0, D = 0, E = 0, and F = 2:

F = (10 * 0 * 0 * 0 * 2 + 1) / (9 * 0 * 0 * 0 * 2 + 1) = (0 + 1) / (0 + 1) = 1 / 1 = 1

Since F = 2, this is not a valid solution.

3. Let's assume B = 0, C = 0, D = 0, E = 0, and F = 3:

F = (10 * 0 * 0 * 0 * 3 + 1) / (9 * 0 * 0 * 0 * 3 + 1) = (0 + 1) / (0 + 1) = 1 / 1 = 1

Since F = 3, this is not a valid solution.

4. Let's assume B = 0, C = 0, D = 0, E = 0, and F = 4:

F = (10 * 0 * 0 * 0 * 4 + 1) / (9 * 0 * 0 * 0 * 4 + 1) = (0 + 1) / (0 + 1) = 1 / 1 = 1

Since F = 4, this is not a valid solution.

5. Let's assume B = 0, C = 0, D = 0, E = 0, and F = 5:

F = (10 * 0 * 0 * 0 * 5 + 1) / (9 * 0 * 0 * 0 * 5 + 1) = (0 + 1) / (0 + 1) = 1 / 1 = 1

Since F = 5, this is not a valid solution.

6. Let's assume B = 0, C = 0, D = 0, E = 0, and F = 6:

F = (10 * 0 * 0 * 0 * 6 + 1) / (9 * 0 * 0 * 0 * 6 + 1) = (0 + 1) / (0 + 1) = 1 / 1 = 1

Since F = 6, this is not a valid solution.

7. Let's assume B = 0, C = 0, D = 0, E = 0, and F = 7:

F = (10 * 0 * 0 * 0 * 7 + 1) / (9 * 0 * 0 * 0 * 7 + 1) = (0 + 1) / (0 + 1) = 1 / 1 = 1

Since F = 7, this is not a valid solution.

8. Let's assume B = 0, C = 0, D = 0, E = 0, and F = 8:

F = (10 * 0 * 0 * 0 * 8 + 1) / (9 * 0 * 0 * 0 * 8 + 1) = (0 + 1) / (0 + 1) = 1 / 1 = 1

Since F = 8, this is not a valid solution.

9. Let's assume B = 0, C = 0, D = 0, E = 0, and F = 9:

F = (10 * 0 * 0 * 0 * 9 + 1) / (9 * 0 * 0 * 0 * 9 + 1) = (0 + 1) / (0 + 1) = 1 / 1 = 1

Since F = 9, this is not a valid solution.

From the above analysis, we can see that the only valid solution is when B = 0, C = 0, D = 0, E = 0, and F = 1. Therefore, the first digit of the original six-digit number is 1.

Answer

The first digit of the six-digit number is 1.

Note: The complete six-digit number cannot be determined without additional information.