Guided Example #1 - Natural Numbers
To construct a 10 digit number that is divisible by 3, 4 and 5, only by using our knowledge of divisibility rules, what must be true about this number?
I. the last two digits must be a number divisible by 4
II. the sum of all the digits must be divisible by 3
III. the last digit must be a zero
IV. the tens digit must be even
V. the ones digit must be a 5 or a 0
Not quite; if we also include IV, this is true.
True; in this case we see in order for all the divisibility rules to hold for 3, 4 and 5, we cannot have a 5 in the ones place, so we must have a 0. Then, in order for the last two digits to be divisible by 4, the tens digit must be even.
Not quite; we also must have II holding in order to have divisibility by 3.
False in this case. The ones digit cannot be a 5 because it will not be divisible by 4 in this case. Instead the ones digit must be a zero. Also the tens digit must be even in order for the last two digits to be divisible by 4.