Well-ordering and division theorem

Reference no: EM13121869

1. Show that there is no rational number b/a whose square is 2, as follows: if b^2 = 2a^2, then b is even, so b = 2c, so, substituting and cancelling 2, 2c^2 = a^2. Use that argument and well-ordering to show that there can be no natural number a > 0 with b^2 = 2a^2 for some natural number b.

2. Let m be the least common multiple of a and b, and let c be a common multiple of a and b. Show that m divides c. Hint: use the division theorem on m and c, and show that the remainder r is a common multiple of a and b, hence r = 0.

Reference no: EM13121869

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