Question

Сумма квадратов цифр задуманного двузначного числа равна 65. Если к задуманному числу прибавить 27, то получиться число, записанное теми же цифрами, что и задуманное число, но в обратном порядке. РЕШИТЬ СИСТЕМОЙ ОБЯЗАТЕЛЬНО.

Answer 1

Задуманное число записано двумя цифрами :
х - цифра десятков и у - цифра единиц
Это число 10х+у
Число записанное этими же цифрами, но в обратном порядке
10у+х
Система
$\left \{ {{10x+y+27=10y+x} \atop { x^{2} + y^{2} =65}} \right. \\ \left \{ {{9x-9y=-27} \atop { x^{2} + y^{2} =65}} \right. \\ \left \{ {{x-y=-3} \atop { x^{2} + y^{2} =65}} \right. \\ \left \{ {{y=x+3} \atop { x^{2} + (x+3)^{2} =65}} \right.$
Решаем второе уравнение
х²+х²+6х+9=65
2х²+6х-56=0
х²+3х-28=0
(х-4)(х+7)=0
х-4=0      или    х+7=0
х=4                    х=-7  не удовлетворяет условию задачи ( цифры от 1 до 9 все положительны)
у=х+3=4+3=7
Ответ  47
Проверка
47+27=74

Answer 2

Problem Analysis

We are given a two-digit number where the sum of its digits squared is equal to 65. When we add 27 to this number, we get a new number that has the same digits but in reverse order. We need to find the original number.

Solution

Let's assume the original number is represented by the tens digit x and the units digit y. According to the problem, we have the following equations:

1. x^2 + y^2 = 65 (Equation 1) 2. 10x + y + 27 = 10y + x (Equation 2)

We can solve this system of equations to find the values of x and y.

Solving the System of Equations

Let's solve Equation 2 for x:

10x + y + 27 = 10y + x

Simplifying, we get:

9x - 9y = 27

Dividing both sides by 9, we have:

x - y = 3 (Equation 3)

Now, let's solve Equation 3 for x:

x = y + 3

Substituting this value of x into Equation 1, we get:

(y + 3)^2 + y^2 = 65

Expanding and simplifying, we have:

2y^2 + 6y - 56 = 0

Dividing both sides by 2, we get:

y^2 + 3y - 28 = 0

Factoring the quadratic equation, we have:

(y + 7)(y - 4) = 0

This gives us two possible values for y:

1. y + 7 = 0 => y = -7 2. y - 4 = 0 => y = 4

Since we are dealing with a two-digit number, we can discard the negative value of y. Therefore, y = 4.

Substituting this value of y back into Equation 3, we can find x:

x = y + 3 = 4 + 3 = 7

Therefore, the original number is 74.

Answer

The original two-digit number is 74.