1 pt Check all the statements that are true A If a and b bot

(1 pt) Check all the statements that are true:

A. If a and b both divide c, then ab divides c².
B. If p and q are distinct primes, then p²q² has exactly 11 positive divisors.
C. If p and q are distinct primes, then p+q is prime as well.
D. If a divides b and c divides d, then a+c divides b+d.
E. If p is prime, then so is p+2.
F. If a and b both divide c, then ab divides c.
G. If a and b both divide c, and a and b are relatively prime, then ab divides c.
H. There are infinitely many prime numbers.
I. If p is prime, then p² has exactly 3 positive divisors.
J. There are three consecutive odd numbers that are prime.

Solution

A. a divides c => c = ar for some intezer r ad similarly b divides c => c=bq for some intezer q

So we get c² = abrq = abt (because rq = t (some intezer))

=> c² divides ab

True

B. False because p²q² has 1,p,q,p²,q²,pq,p²q,pq²,p²q² are all possible divisors and these are exactly 9

C. False because let p=5 and q=7 be distince primes and p+q = 5+7 = 12 which is not a prime

D.False eg. if 3 divides 21 and 5 divides 25 then 3+5 = 8 does not divide 21+25 = 46

E. False.eg p=2 is a prime but 2+2 = 4 is not a prime

F. False. eg. 2 and 12 both divides 36 but 2.12 = 24 does not divide 36

G. True

H.True. Euclid proof

I True because 1,p,p² are only three divisors in this case

J. True 3,5,7 are consecutive odd numbers that are prime