Brain Teasers
Fives
What
is the smallest positive integer that,
when multiplied by 7, yields a product
in which all of the digits are 5’s?
(Can you find the answer without a calculator?)
This brainteaser was
written by Derrick Niederman.
Solution:
79,36
5.
One way to tackle the problem is to find the first number
consisting of all 1’s (called a repunit)
that is divisible by 7. To do that, just set up a standard long division,
adding as many 1’s as you need until you get a remainder of zero. You’ll get
the equation 111,111 = 7 × 15,873, and because this equation is the smallest of
its kind, we can multiply both sides by 5 to yield 555,555 = 7 × 79,365.
You can also multiply backwards to find the answer. That is,
what number times 7 gives a 5 as the units digit in the result? Obviously,
that’s 5. But then 5 × 7 = 35, so a 3 carries to the tens column. So then, what
number times 7, and then adding the carried 3, gives 5? That’s 6, since 6 × 7 +
3 = 45. That gives a 4 that carries… and so on. This process will eventually
lead to the answer, too.
Of course, the method that requires the least amount of
thinking is to just use a calculator. Enter a string of 5’s, divide by 7,
and keep increasing the number of 5’s until the answer is an integer.
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