Math Is Fun Forum

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Why are the last numbers written 09 and 00 rather than 9 and 0? Is that supposed to be a clue?

Hint:

Not necessary, since 1000 is divisible by 8. For any number with four or more digits, you only need to concentrate on the first three digits on the right. Thus 6532 (mod 8) = 532 (mod 8) and 123456789 (mod 8) = 789 (mod 8).

This reason it works is this. Suppose 100a+10b+c is your three-digit number. Doing what you say produces 4a+2b+c. The difference between this and your three-digit number is 96a+8b = 8(12a+b), which is divisible by 8; hence the two numbers are equal mod 8.

Every metric space can be “completed” in a way similar to the construction of the real numbers from the rationals by equivalence classes of Cauchy sequences. http://z8.invisionfree.com/DYK/index.php?showtopic=193

A word about the cancellation law. In a group, the cancellation law applies because every element has an inverse: if a, b, c are elements of a group and ab = ac, we can multiply on the left by the inverse of a to get b = c. This does not apply to elements of a ring under multiplication because multiplicative inverses do not always exist in a ring. If a, b, c are elements of a ring and ab = ac, we cannot conclude that b = c (even if a ≠ 0). Example:

In the ring of 2×2 matrices over the integers, take

In the case of the integers, it is true that ab = ac implies b = c if a ≠ 0. The reason, however, is NOT the cancellation law; the reason is because the integers form an ordered ring. The order axioms are what force the integers to be an integral domain.

I will perhaps discuss more of this in the Euler Avenue section when I have time.