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?

The number is 4789
conditions 1,8, and 9 are met by this number...below is the list of numbers which I found which the sums, sums of squares, and product contain different digits and a certain number of conditions are met

You know that the number must end in 9 or 8, if it were to end in 7 the largest would be 4567 whose digits product is 840 but sum of squares contains 3 digits.  smallest number 1234 whos digit sums and products all contain 2 digits.  If you assume the number to be prime, the number must therefore end in 9.  I found only one example where the number ended in 8 and that number met only 1 condition.

FOR 3 CONDITIONS TRUE
4789 {1,0,0,0,0,0,0,1,1} 28  210  2016

FOR 2 CONDITIONS TRUE
3589 {0,0,1,0,1,0,0,0,0} 25  179  1080
4579 {0,0,0,0,1,0,0,0,1} 25  171  1260
5689 {1,0,0,0,0,0,0,1,0} 28  206  2160

FOR 1 CONDITIONS TRUE
8 numbers meet 1 condition (2789, 3679, 3689, 4678, 4679, 4689, 5679, 5789)

Posted by Christian on 2004-12-23 22:24:39