Buying a house with mickles and bimes

Number Theory Level 1

You just won enough money from a lottery to buy a house! Unfortunately, the money only comes in mickles and bimes...

Mickles are worth $\$ 0.03$ and bimes are worth $\$ 0.07$ .

The problem is that the house you are buying doesn't have a set price, and you must buy the house for the EXACT price.

The price of the house is anywhere between $\$ 111,111.11$ to $\$ 222,222.22$ . Is it true you can guarantee that you can choose a combination of mickles and bimes such that you can purchase the house?

False True

1 solution

Timothy Cao
Jan 21, 2019

Consider the following combinations:

$4 * \$0.03 = \$0.12$

$\$0.07 + 2 * \$0.03 = \$0.13$

$2 * \$0.07 = \$0.14$

Since this is 3 consecutive amounts of cents in a row, we can simply add a multiple of $\$0.03$ to each to achieve any cost greater than $\$0.12$ . Since $\$0.12 \leq \$111,111.11$ , You can definitely buy the house for the exact price!

i.e.

$4 * \$0.03 + \color{#D61F06}{\$0.03 }$ $= \$0.15$

$\$0.07 + 2 * \$0.03 + \color{#D61F06}{\$0.03 }$ $= \$0.16$

$2 * \$0.07 + \color{#D61F06}{\$0.03 }$ $= \$0.17$

Just a note that your question is a yes or no, but your choices of answer are true/false.

Jeremy Galvagni - 2 years, 4 months ago

Thank you, fixed.

Timothy Cao - 2 years, 4 months ago

0.07+0.03=0.10 0.10*10=1

Nishant Sahoo - 1 year, 1 month ago

Buying a house with mickles and bimes

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