### Video Transcript

Find the smallest number which is
divisible by both five and two and is formed from five different digits.

As the number needs to have five
digits, it must have a number in the ten thousands column, the thousands column, the
hundreds column, the tens column, and the units column. We are also told that the five
digits need to be different. This means that one possible number
is 98765. This is the largest possible number
that can be formed from five different digits.

Our number cannot start with a
zero. This means that the smallest
possible number that can be formed from five different digits is 12340. We have used the digits zero, one,
two, three, and four. In this question, we need a number
which is divisible by both five and two. Any whole number is divisible by
five if its units digit is five or zero. Any whole number is divisible by
two if it is even. This means that it needs to end in
either two, four, six, eight, or zero.

As we want a number that is
divisible by both five and two, the units digit of our five-digit number must be
zero. As we want the smallest number,
let’s consider the number 12340. This was the smallest possible
number formed from five different digits. As our number ends in zero, there
was a zero in the units column, we know it is divisible by five. It is also an even number. Therefore, it is divisible by
two. The smallest number which is
divisible both by five and two and is formed from five different digits is
12340.

Once we know that this number is
divisible by both five and two, we could check this using the short division bus
stop method. Let’s firstly divide 12340 by
five. Five does not divide into one. Therefore, we need to carry the one
to the next column. 12 divided by five is equal to two
remainder two. 23 divided by five is equal to four
remainder three. We have to carry the remainder
again. 34 divided by five is equal to six
remainder four. And finally, 40 divided by five is
equal to eight. 12340 divided by five is equal to
2468.

Next, we can check 12340 divided by
two. One is not divisible by two, so we
need to carry the one to the thousands column. 12 divided by two is equal to
six. Three divided by two is equal to
one remainder one. 14 divided by two is equal to
seven. And finally, zero divided by two is
equal to zero. 12340 divided by two is equal to
6170. We have, once again, proved that
12340 is divisible by five and two.