Question Video: Finding the Greatest or Smallest 𝑛-Digit Number That Is Divisible by the Given Numbers

Find the smallest number which is divisible by both 5 and 2 and is formed from 5 different digits.

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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.

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