Question

[Trịnh Quỳnh Hoa]

Exer 1: Given two natural numbers whose sum are 78293. The bigger number where 5 is the units digit and 2 is hundred digit. If we clean these digits then we obtain a number which equals the smaller number. Find two natural numbers.

Exer 2: Prove that: If x, y \(\in\) N and x + 2y divisible by 5 then 3x - 4y divisibles by 5.

Exer 3: Given that 2x + 5y \(⋮\) 7. Prove that 4x + 3y \(⋮\) 7.

Answer 1

[Aoko-chan]

Exer 1:

Solution:

Suppose that, the unknown number is: \(\overline{x215}\) (where x \(\in\) N).

When we clean three digits then the smaller number is \(\overline{x}\).

We have: \(\overline{x215}\) + \(\overline{x}\) = 78293

\(\Rightarrow\) 1000. \(\overline{x}\) + 215 + \(\overline{x}\) = 78293

1001. \(\overline{x}\) = 78078

x = 78

Thus, we found two natural number: 78215 and 78.

Exer 2:

Solution:

We have: x + 2y \(⋮\) 5

\(\Rightarrow\) 2x + 4y \(⋮\) 5

(2x + 4y) + (3x - 4y) = 5x \(⋮\) 5

\(\Rightarrow\) 2x + 4y \(⋮\) 5

Deduce 3x - 4y \(⋮\) 5.

Exer 3:

Solution:

We have: 2x + 5y \(⋮\) 7

4x + 10y \(⋮\) 7

(4x + 10y) - (4x + 3y) = 7y \(⋮\) 7

\(\Rightarrow\) 4x + 10y \(⋮\) 7

Deduce 4x + 3y \(⋮\) 7.