Elementary School Level Number Theory

學術區
1

Question:

Prove that 0.\dot{9} = 1 without using calculus.

2

Solution:

By definition, 0.\dot{3} = 1/3
Then 0.\dot{9} = 3 x 0.\dot{3} = 3 x 1/3 = 1

3

Question:

Prove that 999999 is divisible by 7.

4

Solution:

Notice the last digits, as we know the following patterns:

XXX1 x 9 = XXX9
XXX3 x 3 = XXX9
XXX7 x 7 = XXX9

for whatever digits X.

Now, by distributive law of numbers, we have:

142857 x 7 = 999999