Number Theory, HCF and LCM

The largest known prime number as of 2024 is a Mersenne prime of the form 2^82,589,933 − 1, not '282,589,933 − 1'. The options given are incorrect representations of the exponent or the formula. Therefore, 'None of these' is the correct choice.

Zero is classified as an even number because it is divisible by 2 without any remainder (0 ÷ 2 = 0). It is also a whole number, as whole numbers include zero and all positive integers. However, since the question asks for a single classification and zero is definitively even by the standard mathematical definition, the most precise answer is that zero is an even number.

The series is: 4, 8, 9, 13, 14, 18, ... The pattern alternates between adding 4 and adding 1: 4 + 4 = 8, 8 + 1 = 9, 9 + 4 = 13, 13 + 1 = 14, 14 + 4 = 18. Following this pattern, the next step is to add 1 to 18, resulting in 19. Therefore, the correct next number in the series is 19, which corresponds to option D.

The pattern in the sequence is that each number is multiplied by 2 and then 1 is added: 16×2+1=33, 33×2-1=65, 65×2+1=131, 131×2-1=261, 261×2+1=523. Thus, the missing number is 523.

The pattern in the sequence is: add 3, 5, 7, 9, 11, 13, and 15 to each previous number. The last number should be 65 (50 + 15), not 64. Therefore, 64 is the odd one out.