Question

1 . Care zerouri sare la sfarsit  numarul : 1 x 2 x 3 x 4 x . . .  x 50 ?

2 . Stabileste daca :
a ) 9^7 si 4^5 sunt patrate perfecte  ;
b ) ( 7^4 + 7^5 ) x 49 este cub perfect ;

Answer 1

$1) 1*2*3*4*5*...*50 \\\\ 2*5=10 \to 1 \ zero \\ 4*25=100 \to 2 \ zero-uri \\6*35=210\to 1 \ zero \\ 8*45=360\to1 \ zero \\15*12=180\to 1 \ zero\\ 10 \\ 20 \\ 30 \\ 40 \\ 50*14=700\to2 \ zero-uri \\\\ \boxed{ 12 \ zero-uri \ in \ total}$

$2) \\\\ 9^1=(....9) \\ 9^2=(....1) \\ 9^3=(....9) \\\\ Se \ repeta \ din \ 2 \ in \ 2 =>9^7=(....9) -este \ p.p \\\\\\ 4^1=(....4) \\ 4^2=(....6) \\ 4^3=(....4) \\\\ Se \ repeta \ din \ 2 \ in \ 2 => 4^5=(....4)-este \ p.p$

$3) \\\\ (7^4+7^5)*49= \\\\ =7^4(1+7)*7^2= \\\\ = 7^4*8*7^2= \\\\ = 7^6*8= \\\\= (7^2)^3*8= \\\\ = 49^3*2^3\to este \ cub \ perfect$