## Product and Sum

Arithmetic, algebra, number theory, sequence and series, analysis, ...
Posts: 8
Joined: Sat Apr 03, 2010 6:20 am

### Product and Sum

Fairly easy problem, but interesting to see...

Find the only 4-digit integer which the product of its digits is equal to the first two digits of the number, and the sum of its digits is equal to the last two digits of the number. Or,
Let the 4-digit number be abcd
a.b.c.d = ab
a+b+c+d = cd
Where abcd, ab, and cd are integers and a.b.c.d is product of digits.

If you can answer it correctly, can you find all the 5-digit integers which has similar property, the product of its digits is equal to the first three digits of the number, and the sum of its digits is equal to the last two digits of the number?

torrocus
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Joined: Sat Apr 03, 2010 8:16 am
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### Re: Product and Sum

It's only two the 5-digit integers which has similar property.
I also found one 6-digit integers (product of the first four digits and sum of the last two digits).

uws8505
Posts: 58
Joined: Tue Sep 30, 2008 3:13 pm
Location: South Korea

### Re: Product and Sum

Possible approach:
a * b * c * d = 10a + b
a + b + c + d = 10c + d
Since 10a + b is multiple of a, b is also multiple of a.
a + b + c + d = 10c + d gives a + b = 9c.
The only possible values for c are 1 and 2, but if c = 2, then a = b = 9 and it is impossible. (99 = 9 * 11)
So c = 1, a + b = 9.
b is multiple of a and a + b = 9, so possible values for a and b are (1, 8) and (3, 6).
However, 18 is not a multiple of 8, so a = 3, b = 6, c = 1, d = 2.
The answer for 4-digit problem is 3612.
Math and Programming are complements