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 Math Class (Posted on 2004-12-23)
The math teacher addressed the class. "I have chosen a positive integer (base 10). It has 4 digits, in ascending sequence, none of which are 0. I have calculated the sum of its digits, the sum of the squares of its digits, and the product of its digits, and each of these quantities has a different number of digits. Consider the following 9 statements:
(1) The number is a prime.
(2) The sum of its digits is a prime.
(3) The sum of the squares of its digits is a prime.
(4) The number is a square.
(5) The sum of its digits is a square.
(6) The sum of the squares of its digits is a square.
(7) The number is triangular.
(8) The sum of its digits is triangular.
(9) The sum of the squares of its digits is triangular.

You must use them to determine the number."
"But they can't all be true!", interjected one of the students.
"I never said they were! Some of the statements are true and some are false."
"Well we will need more information. Tell us which are true and which are false."
"If I told you that you would easily be able to determine the number!"
"Well at least tell us how many are true."
"If I told you that now, you would be able to determine the number too easily!"
"Well, what more can you tell us?"
"Nothing! I have told you enough!"

What was the number?

 See The Solution Submitted by Jim Rating: 3.6667 (6 votes)

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 Solution | Comment 3 of 9 |
The solution is 4789

There are 126 numbers with 4 digits in ascending order. Only 19 of these have different numbers of digits for the digit sum, digit square sum and product sum.

For each I found how many of the statements they made true.
7 are true for none of them
7 are true for one of them
2 are true for two of them and only
1 is true for three of them.

This number is 4789
It has digit sum = 28, digit squared sum = 210 and digit priduct = 2016 (each a different # of digits)
It is prime (1)
The sum of its digits is triangular (8)
The sum of the squares of its digits is triangular (9)

-Jer
 Posted by Jer on 2004-12-23 22:03:43

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