If the 5 digit number #1364"?"# is divisible by #3# then what are the possible values of the last digit?

Question

Everly Taylor · Accepted Answer

#1#, #4# or #7#.

In the decimal number system that we use, an integer is divisible by #3# if and only if the sum of its digits is also divisible by #3#. #1+3+6+4+color(blue)(1) = 15# is divisible by #3#. So #1364color(blue)(1)# is divisible by #3#, and so will be: #1364color(blue)(1)+3 = 1364color(blue)(4)# and #1364color(blue)(4)+3 = 1364color(blue)(7)# #color(white)()# Footnote Why does this method of checking the digits add up to a multiple of #3# work? Essentially because when you divide #10# by #3# then the remainder is #1#. So for example: #color(red)(153) = (100*color(red)(1)) + (10*color(red)(5)) + (1*color(red)(3))# #= (99+1)*color(red)(1) + (9+1)*color(red)(5) + (0+1)*color(red)(3)# #= (99*color(red)(1) + 9*color(red)(5) + 0*color(red)(3)) + (1*color(red)(1) + 1*color(red)(5) + 1*color(red)(3))# #= 3(33*color(red)(1) + 3*color(red)(5) + 0*color(red)(3)) + (color(red)(1)+color(red)(5)+color(red)(3))# The expression #3(33*color(red)(1)+3*color(red)(5)+0*color(red)(3))# is divisible by #3#. So we find that #153# is divisible by #3# if and only if #(1+5+3)# is.

Everly Taylor · Answer

undefined To find the possible values of the last digit of the 5-digit number 1364?, such that it is divisible by 3, we need to consider the divisibility rule for 3. The divisibility rule for 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3.

The sum of the digits in the number 1364? is $1 + 3 + 6 + 4 + ? = 14 + ?$. For the number to be divisible by 3, the sum must also be divisible by 3. Therefore, we need to find the values of "?" such that $14 + ?$ is divisible by 3.

We can try different values of "?" to see which ones make the sum divisible by 3.

If we start from "?" being 0, then the sum is 14. Since 14 is not divisible by 3, "?" cannot be 0.

If we try "?" as 1, then the sum becomes 15, which is divisible by 3.

If we try "?" as 2, then the sum becomes 16, which is not divisible by 3.

If we try "?" as 3, then the sum becomes 17, which is not divisible by 3.

If we try "?" as 4, then the sum becomes 18, which is divisible by 3.

Continuing this pattern, we find that the possible values for "?" are 1 and 4. Therefore, the possible values of the last digit of the number 1364? such that it is divisible by 3 are 1 and 4.