find the even number less than 100 that is divisible by 5 and when divided by 9 has a remainder of 8

Question

find the even number less than 100 that is divisible by 5 and when divided by 9 has a remainder of 8

Answer 1

You can learn a lot by just listing multiples of a number:

 

A:  “Even numbers” (multiples of 2) are:

     2, 4, 6, 8, 10, 12, 14, 16, 18, …    (end with 0, 2, 4, 8 ,…repeated)

 

B:  “divisible by 5” are:

    5, 10, 15, 20, … (end with 0,5, …. Repeated)

 

   “divisible by 9” are:

   9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99… (last digit 9,8,7,6,5,4,3,2,1,0 repeated)

C: “divided by 9 with remainder 8” are:

   8, 17, 26, 35, 44, 53, 62, 71, 80, 89, 98,,, (last digit 8,7,6,5,4,3,2,1,0,9 repeated)

 

You can tell immediately that 80 is the only number that fits A and B and C.

 

Now, number theory explains patters in math terms.  However, it is not too hard to list all the numbers between 1 and 1000 that satisfy A, B and C above:

  80  170  260  350  440  530  620  710  800  890  980   (see a pattern??)

 

You could list all the numbers between 1 and 10,000 that satisfy A, B and C above in about a minute.

80  170  260  350  440  530  620  710  800  890  980

1070  1160  1250  1340  1430  1520  1610  1700  1790  1880  1970
2060  2150  2240  2330  2420  2510  2600  2690  2780  2870  2960
3050  3140  3230  3320  3410  3500  3590  3680  3770  3860  3950
4040  4130  4220  4310  4400  4490  4580  4670  4760  4850  4940
5030  5120  5210  5300  5390  5480  5570  5660  5750  5840  5930
6020  6110  6200  6290  6380  6470  6560  6650  6740  6830  6920
7010  7100  7190  7280  7370  7460  7550  7640  7730  7820  7910
8000  8090  8180  8270  8360  8450  8540  8630  8720  8810  8900
8990  9080  9170  9260  9350  9440  9530  9620  9710  9800  9890
9980

 

Hint:  list all the 0's at the end for all the numbers, then list 8,7,6,5,4,3,2,1,0,9 repeated for the next to last number, then ... (well, you take it from here)

Answer 2

 

Even numbers end with 0, 2, 4, 6. or 8.

Numbers divisible by 5 end with 0 or 5.

 

You need the remainder to be 8 when the number is divided by 9.

 

Since the number is less than 100.  your choices are:

 

0, 10, 20, 30, 40, 50, 60, 70, 80, and 90.

 

divide each by 9 ... the one that has a remainder of 8, is your number.

Answer 3

Debbie,

 

An even number is divisible by 2. If it is also divisible by 5, it is then divisible by 10. If you divide a multiple of 10 by 9 and get a remainder of 8, then the number must be of the form

 

n = 9a + 8 and also of the form n = 10b.

 

If you set these two equations equal to each other, you will get

 

9a + 8 = 10b, 

 

or

 

9a - 10b = -8.

 

One way to solve this would be to try every multiple of 10 less than 100, divide it by 9, and look at the remainder. I do not know how much math you know at this point, but this is a typical question in number theory. If you had to find a number less than 1000 or a large number, using trial and error would not be practical.

 

Can you take it from here?

 

I will tell you that the number is larger than 50, but you still have to find it.

 

Good luck!