Question

Show that the product of two of the numbers 65^1000 − 8^2001 + 3^177, 79^1212 − 9^2399 + 2^2001, and 24^4493 − 5^8192 + 7^1777 is nonnegative. Is your proof constructive or nonconstructive? [Hint: Do not try to evaluate these numbers!]

Answer 1

To show that the product of two of the given numbers is nonnegative, we need to prove that either both numbers are nonnegative or both numbers are negative.

Step-by-step explanation:

To show that the product of two of the given numbers is nonnegative, we need to prove that either both numbers are nonnegative or both numbers are negative. Let's consider each pair:

1. 65^1000 − 8^2001 + 3^177 and 79^1212 − 9^2399 + 2^2001: Both numbers have the same sign because the exponents of 2 and 8 are even. Therefore, their product is nonnegative.
2. 65^1000 − 8^2001 + 3^177 and 24^4493 − 5^8192 + 7^1777: Again, both numbers have the same sign because the exponents of 2, 5, and 8 are even. Therefore, their product is nonnegative.
3. 79^1212 − 9^2399 + 2^2001 and 24^4493 − 5^8192 + 7^1777: The exponents of 2 and 5 are odd in both numbers, so they have the same sign. Therefore, their product is nonnegative.

Hence, the product of any two numbers among the given three is nonnegative.