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1. Suppose n ≡ 7 (mod 8). Show that n ≠ x2 + y2 + z2 for any x, y, z ε Z.
2. Prove ∀n ε Z, that n is divisible by 9 if and only if the sum of its digits is divisible by 9.
3. Prove that it is always possible to make postage of exactly n cents for all n ≥ 32 using only 5 and 9 cent stamps.
4. Prove that every fourth Fibonacci number is a multiple of 3.
In other words, show that 3 | f4n ∀n ≥ 1.
5. Let bn be the sequence recursively defined by b0 = 1, b1 = 5 and, for n > 1,
bn = b[n/3]+2b[n/3]
(a) Compute b26 and b27.
(b) Guess a formula for bn when n = 3t for t ≥ 0 and then use mathematical induction to prove that your guess is correct. (Be sure to include a careful statement of what you are trying to prove).
6x^7-2x^3+4x-16 / 3x^2-7x+9
Describe and sketch the surfaces z + |y| = 1 and (x 2) 2 y + z 2 = 0.
how to determine roman numerals to digits specially when it hundred thousands
Solve the fractional equation: Example: Solve the fractional equation 1/(x-2) +1/(x+3) =0 Solution: The LCD is (x - 2)(x + 3); therefore, multiply both sides of t
how to see shares and dividends of a company and are they seen day wise?
Demerits and merits of the measures of central tendency The arithmetic mean or a.m Merits i. It employs all the observations given ii. This is a very useful
f(x)=sin x+cos x in the interval {0,90}
HOW MANY ZERO ARE THERE AT THE END OF 200
At rest, the human heart beats once every second. At the strongest part of the beat, a person's blood pressure peaks at 120mmHg. At the most relaxed part of the beat, a person's bl
lnx(1+x)
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