This problem solving activity has a number and algebra (patterns and relationships) focus.
Jim has ten tiles with a different digit on each of them.
He makes a lot of five-digit numbers that are divisible by five.
How many can he make altogether?
This problem investigates numbers with a given factor. Students must first be able to identify the property of a number to determine its factors. For example: a number is divisible by 5 if its last digit is 0 or 5. They must then apply a systematic approach to count in an efficient manner all the numbers with this property, ensuring that none has been missed.
Related problems include: Ten Tiles III, Number and Algebra (Patterns and Relationships), Level 6.
Note: At The Movies, Level 3, Number may be a useful starting place for this problem.
Jim has ten tiles with a different digit on each of them. He makes a lot of five-digit numbers that are divisible by five. How many can he make altogether?
Using the Jim’s tiles, how many four-digit numbers are there that are divisible by four?
For a number to be divisible by five, it has to end with a 5 or a 0. So we have two choices for the last digit. If 0 is the last digit, then there are 9 choices for the first digit of the five-digit number, 8 for the second digit, 7 for the third and 6 for the fourth. So altogether we have 9 x 8 x 7 x 6 = 3024 five-digit numbers that are divisible by five with a 0 in the units position. On the other hand, if 5 is the last digit, there are 8 possibilities for the first digit (all ten digits less the 5 and the 0), then 8 for the second (0 is allowed here), 7 for the third and 6 for the fourth to give a total of 8 x 8 x 7 x 6 = 2688.
Altogether there are 3024 + 2688 = 5712 five-digit numbers divisible by five.
(Again this can be approached by a subtraction method – see Ten Tiles I.)
Four-digit numbers divisible by four: Here the rule is that the number is divisible by 4 if the last two digits are divisible by four. In total there are 25 potential last two digits that are divisible by four but three of these (00, 44 and 88) repeat a digit and so cannot be used here. There are 6 endings that have a 0 (04, 08, 20, 40, 60, 80). These give rise to 8 x 7 numbers each. There are 16 endings that do not have a 0 (12, 16, 24, 28, 32, 36, 48, 52, 56, 64, 68, 72, 76, 84, 92, 86) and these give rise to 7 x 7 numbers. Altogether then there are 6 x 8 x 7 + 16 x 7 x 7 = 1120 four-digit numbers divisible by four.
Printed from https://nzmaths.co.nz/resource/ten-tiles-ii at 4:32am on the 20th April 2024