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jmc

number theory intermediate

Problem

If

◊ 4_{7} = ◊ 1_{8}

and

◊

represents a digit, solve for

◊

.

Solution

Converting everything to base ten:

◊ 4_{7} ◊ \cdot 7^{1} + 4 \cdot 7^{0} 7 ◊ + 4 ◊ = ◊ 1_{8} = ◊ \cdot 8^{1} + 1 \cdot 8^{0} = 8 ◊ + 1 = 3 .

Final answer

3