Base System

Rarely Tested

A number base is the number of unique digits, including zero, used to represent numbers.

For Ex: Binary System = Base 2 (0,1), Octal System = Base 8 (1-7),  Decimal System = Base 10 (1-9)

Any number in base b can be written like this:$$(d_n​d_{n-1}​\dots d_1​d_0​.d​_{-1}d_{-2}​\dots)_b​$$

                            Where: $$(d_i$$ are the digits (each must be less than b)

To convert it to decimal (base 10): Multiply each digit by b raised to its position: $$d_nb^n+d_{n-1}b^{n-1}+\dots+d_1b^1+d_0b^0+d_{-1}b^{-1}+d_{-2}b^{-2}+\dots$$

Question 1

A natural number X (less than 100) is represented in base notation 2, 3 and 5. In each of the three representations, the last digit is 1 while the first digit is 1 in only one of the three notations. What is X?

Question 2

X is the smallest positive integer greater than 100 such that, if its leftmost digit is deleted, the resulting integer is $$\frac{X}{19}$$. Which of the following statements about X is false?

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