Prove that 16 divides n4 + 4n2 + 11, if n is an odd integer.

Q1 of 128 Page 4

Prove that 16 divides n⁴ + 4n² + 11, if n is an odd integer.

Given here, n is an odd integer for some k Z

where Z is the set of all integers.

Since, we know that every odd integer is of the form 4k + 1 and 4k – 1.

Consider two cases:

Case 1: For n = 4k + 1

)

Therefore, it is divisible by 16.

Case 2: For n = 4k – 1

)

Therefore, it is divisible by 16.

Thus, n⁴ + 4n² + 11 is divisible by 16.

Prove that if n is a positive even integer, then 24 divides n(n + 1)(n + 2).

Prove that if either of 2a + 3b and 9a + 5b is divisible by 17, so is the other. a, b E N (Hint: 4(2a + 3b) + 9a + 5b = 17a + 17b)

Prove that every natural number can be written in the form 5k or 5k ± 1 or 5k ± 2, k E N V {0}.

Prove that if 6 has no common factor with n, n² — 1 is divisible by 6.