Question

1.1 К записи натурального числа слева и справа приписали цифру 1. в результате число увеличилось в 21 раз. найдите самое маленькое из таких чисел. 1.2 Сумма двух чисел равна 136, если у одного из них зачеркнуть цифру равную 4, то получится второе число. Какие это числа? 1.3 Сумма номеров домов на одной стороне квартала равна 333. Определите номер дома, пятого от угла квартала.

Answer 1

1.1    Имеем   x = 10a+b ⇒ y = 1000+100a+10b+1 = 1001+10(10a+b)= 1001+10x 
              ⇒  (1001+10x)/x = 21  
                     1001+10x = 21x
                      11x= 1001
                           x=91
1.2     Очевидно, что  один  из  чисел  трехзначное( 100a+10b+c)  и
          заканчивается  цифрой 4 (c=4)  и  т.к.  сумма  136  ⇒ a=1 .
            Второе  число  должен  заканчиваться  цифрой  2 .
           Т.к.  после  зачеркивания  цифры  4,  числа  равняться  ⇒ b=2
                   ⇒  первое  число  124,  второе  12 

Answer 2

1.1 Smallest Number with a Digit Added on Both Sides

To find the smallest number that, when a digit '1' is added to both its left and right sides, increases by 21 times, we can set up the equation as follows:

Let the original number be represented as "x". When '1' is added to both sides, the new number becomes "1x1".

According to the given condition, the new number is 21 times the original number: 1x1 = 21x

To solve for "x", we can use the following steps: 1x1 = 21x 1(10a + x + 1) = 21(10x + a) 10a + x + 1 = 210x + 21a 10a - 21a = 210x - x - 1 -11a = 209x - 1 x = (-11a + 1) / 209

By trying different values for "a", we can find the smallest integer value for "x" that satisfies the condition.

The smallest number that satisfies the given condition is x = 10.

[[1 #6]]

1.2 Finding Two Numbers with a Specific Property

Given that the sum of two numbers is 136 and one of the numbers can be obtained by removing the digit 4 from the other number, we can set up the following equations:

Let the two numbers be represented as "ab4" and "cd".

According to the given condition, the sum of the two numbers is 136: ab4 + cd = 136

Also, removing the digit 4 from the first number gives the second number: ab - 4 = cd

By solving these equations simultaneously, we can find the values of "a", "b", "c", and "d" that satisfy the conditions.

After solving the equations, we find that the numbers are: 104 and 32.

[[2 #9]]

1.3 Determining the House Number

If the sum of the house numbers on one side of the block is 333, and we need to determine the number of the fifth house from the corner, we can set up the following equation:

Let the house numbers be represented as "a", "b", "c", "d", and "e".

According to the given condition, the sum of the house numbers is 333: a + b + c + d + e = 333

To find the number of the fifth house from the corner, we can use the sum of the first four house numbers to determine the fifth house number: a + b + c + d = 333 - e

By substituting the values of "a", "b", "c", and "d" into the equation, we can find the number of the fifth house from the corner.

After solving the equation, we find that the number of the fifth house from the corner is: 79.

[[3 #10]]