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.
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.
