Find A Three Digit Number

Here’s a Number Theory Question:

[A three digit number N satisfies the following condition: N is equal to 5 times the product of digits N. What is N?]

A Solution:

Let N = 100a + 10b + c, where a,b & c are digits between 0 to 9.

Then, the problem requires

100a + 10b + c = 5abc… (1)
So,

c = 5abc – 100a – 10b
c = 5 (abc – 20a – 2b)
c = 5 x (an integer)
c = “a multiple of 5”

The only multiples of 5 between the digits from 0 to 9 are 0 and 5.

We consider each case separately as follows:

Case 1 (c = 0)

Equation (1) becomes,
100a + 10b = 0 which is only possible if both a = b = 0 too. Thus, “no” solutions are possible here.

Case 2 (c = 5)

Then equation (1) becomes

100a + 10b + 5 = 5 x ab x 5
20a + 2b + 1 = 5ab
2b + 1 = 5ab – 20a
2b + 1 = 5 (ab – 4a)… (2)
2b + 1 = 5 x (“an integer”)
2b + 1 = “a multiple of 5″… (3)

But since highest possible value for b is 9, we have
(2b + 1) <= 2(9) + 1 = 19, so
(2b + 1) <= 19… (4)

Now, (3) & (4) tells us that (2b + 1) is a “multiple of 5 which is less than 19”. Possibilities are: 5, 10 or 15 which corresponds to b = 2, 9/2 (not integer, reject!) and 7.

So, b = 2 or 7. Put these possible values back into equation (2) and solve to get:

If b = 2, then
2(2) + 1 = 5 (2a- 4a)
5 = 5 (2a – 4a)
-2a = 1
a = -1/2 (not an acceptable value, reject!)

Thus, it must be that b = 7. Put this in equation (2) and solve to get:

2(7) + 1 = 5 (7a – 4a)
15 = 5 (3a)
3 = 3a
a = 1

Thus, the only possible solution as shown above is:

(a,b,c) = (1,7,5) which corresponds to the three digit number N = 175 (Answer).

Check: 175 = 5 x 1 x 7 x 5 (which is true), End.