A four-digit number is divided by 35 and has a remainder of 15. The first three digits of the number are 155. What is the fourth digit of the number?

We are given
a four-digit number with leading digits 1, 5, and 5. When this number is divided by 35 the
remainder is 15. We are asked to find the last digit.

(1) We can try a guess
and revise strategy. We are looking for a multiple of 35 that is 15 below the number 155x, where
x is an unknown number.

35*25=875 This number is way too low. (25 was just
picked randomly.)

35*40=1400

35*50=1750

35*45=1575 This number is a little too high.

35*44=1540.

If we add 15 to 1540 we get 1555. The first three digits are correct and:

`1555 div 35=44 "R" 15` as required. The last digit is
5.

(2) We can try long division:

35 does
not go into 15 (or 35 does not divide 15.)

35 goes into 155 4 times with
35*4=140. We subtract 140 from 155 to get 15 and we bring down the "x" to get
15x.

Now 35*4=140 so we want 15x-140=15 so the unknown digit is
5.

3) We could list all of the multiples of 35 and add 15 to
them.

...

35*43+15=1520

35*44+15=1555

35*45+15=1590

The middle one has
the correct form.

Why add 15? If a number is a multiple of 35, when we divide
by 35 we get a remainder of 0. If a number is 15 more than a multiple of 35 the remainder after
dividing by 35 will be 15.

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