Question

1.
Află perechile de numere naturale (a, b) care verifică egalitatea:
a x b= 3 + 5+7 +9+ ..... +55 + 57 -2-4-6-..... - 56.​

Answer 1

Răspuns:

Explicație pas cu pas:

a x b= 3 + 5+7 +9+ ..... +55 + 57 -2-4-6-..... - 56.​

axb = (3-2)+(5-4)+(7-6)+.....(57-56) = 1·28 = 28

nr. de perechi de termeni = (57-3):2+1 = 28

a x b = 28 => a = 1 ; b = 28 => (1 ;28)

a = 2 ; b = 14 => (2 ; 14) ; a = 4 ; b = 7 => (4 ; 7)

a = 7 ; b = 4 => (7 ; 4) ; a = 14 ; b = 2 => (14 ; 2)

a = 28 ; b = 1 => (28 ; 1)

Answer 2

$a\times b = 3+5+7+9+...+55+57 - 2-4-6-...-56 \\ \\ a\times b = (2\cdot 1+1)+(2\cdot 2+1)+...+(2\cdot 28+1)- 2\cdot 1-2\cdot 2-...-2\cdot 28\\ \\ a\times b = 1+1+1+\underbrace{...}\limits_{de\,\, 28\,\, ori}+1 \\ \\ a\times b = 1\times 28 \\ a\times b = 2\times 14 \\ a\times b = 4 \times 7 \\ \\ \\ \Rightarrow (a,b)= \Big\{(1,28);(28,1);(2,14);(14,2);(4,7);(7,4)\Big\}$