This proof uses modular arithmetic. If you are unfamiliar with this, the basic principle is that if we have integers a, b, and a nonzero integer c, then a = b (mod c) if a/c and b/c have the same remainder. For example, 8 = 2 (mod 3), because 8/3 and 2/3 have remainder 2.
One property of this relation is that for any integer x and for any nonzero integer y, there exists a unique integer z such that x = z (mod y) and z is between 0 and y inclusive. The upshot of this is that, in most cases, if you know how your relation behaves with all integers between 0 and y, you know how it behaves for all integers.
Consider the quadratic residues mod 8; that is, find all possible values of c if 0 <= c < 8 and x2 = c (mod 8) for some integer x. Plugging in all values from 0 to 7 for x, the only possible values of c are 0, 1, and 4.
Now consider 2n + 1 (mod 8). We know that 2n + 1 is a perfect square, so we know that 2n + 1 = 0, 1, or 4 (mod 8). Thus, 2n = -1, 0, or 3 (mod 8). Since 2n is an even number, and 8 is an even number, 2n can only be congruent to an even number mod 8. Therefore, 2n = 0 (mod 8), and therefore n = 0 (mod 4).
Finally, consider 3n + 1 (mod 8). As before, we note that 3n = -1, 0, or 3 (mod 8). We know that n = 0 (mod 4), so we know that n = 4k for some integer k. Therefore, n is even. Since 8 is also even, we know that n = 0 (mod 8). Therefore, n is divisible by 8. QED.
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Yes. Every perfect square has two roots: one positive and one negative.
8 is, because it's the only perfect cube. 39 is, because it's the only one divisible by 13. 51 is, because it's the only one divisible by 17. 123 is, because it's the only one divisible by 41. 124 is, because it's the only one divisible by 31 892 is, because it's the only one divisible by 223.
Squaring is the function used when we multiply a number by itself. The number you are multiplying is called the baseand the exponent 2 indicates you are multiplying the base by itself. base2 = base x base = square Squaring is the function used when we multiply a number by itself. The number you are multiplying is called the base and the exponent 2indicates you are multiplying the base by itself. base2 = base x base = square
Paul learns that adults are not perfect. He learns that perfect and fair do not always exist. He also learns how he lost his sight.
the answer is 144, it is divisible by 1, 4, 9, 16, 36, and 144.
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The answer is 19.
It is a natural number. It is a positive integer. It is a positive rational number. It is a positive real number. It is a perfect square. It is a three digit integer. It is a palindromic integer. Probably many other sorts.
a perfect number is defined as a positive integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself 137,438,691,328 is the 7th perfect number.
6.
324
Only if the integer is a perfect square.
10
Irrational. The square root of a positive integer is either an integer (that is, if the integer is a perfect square), or an irrational number.
Yes. The square root of a positive integer can only be an integer (if your integer is a perfect square), or an irrational number (if it isn't).
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