Question

(1 point) If a statement has an example that proves it true, please enter it. If instead a statement is able to be disproved, please enter 'F' for false. If you enter an example, please use the one with the smallest possible absolute value. For instance, if 3 and -1 are both counter examples, then the correct answer is -1. 1. 3x E Z(Prime(12.2 +21)) 2. Fx e R \ {0} Vy e R(xy = x) 3. There is a positive prime cube. 4. In EN-Prime(22" + 1)) 5. There are two integers of opposite parity whose product is odd.

Answer 1

1. VKGZ 12x²+21= 3 (4277). => 31 120+21 [:eft is an integere] 7 12 & 721 is not prime 1: 12472173, txE2]. 7 xEZ s.2 12x²+21 is prime. . . 1. is False 2 Take, x=1., then x = 9 2 1= x, & YER. se, 3x21 GIR1Yo}, such that & YELR, x = x. & Take, nal. Then, 2² + 1 = ² + 1 = 4+1=5, is a prime I n=1 EIN such that 22 41 is prime. 5. Let, a and 6 be two integers with opposite parity. Then one must be add and other must be even. Let, a be odd and b be even, ie, a arkt 1 and b2am , where K, MEZ.

Now, a.ba (2x+1), 2m 2 2. (24+1 m, which is multiple of 2. Then aub is even Product of integers with opposite parity is always even. ... The statement 5. is false.