there is a number which is very peculiar.this number is 3 times the sum of its digits.can u find the number?
2007-01-09 00:37:11 · 10 answers · asked by shreya i 2 in Science & Mathematics ➔ Mathematics
27
2007-01-09 00:44:57 · answer #1 · answered by deflagrated 4 · 1⤊ 0⤋
This old chestnut comes back again after 6 months!!!
To answer the question, 27 is the only positive integer with this property. To prove this, let a_k a_(k-1) ... a_1 a_0 be the digits of the integer written in base 10 (so it is a k+1 digit number, and I am assuming that a_k is not 0). Then 3 times the sum of the digits is
3a_k + 3a_(k-1) ... 3a_1 + 3a_0, while the number itself is given by a_k*10^k + a_(k-1)*10^(k-1) + ... +a_1*10^1 + a_0. If these expressions are equal, we have
3a_k + ... + 3a_0 = a_k*10^k + ... + a_0, so that
(10^k-3)a_k = -(10^(k-1) - 3)a_(k-1) - ... - (10-3)a_1 + 2a_0
Now clearly each of the terms (10^i-3)a_i is nonnegative for i running between 1 and k. Thus, the sum on the right is less than or equal to 2a_0, so that (10^k-3)a_k <= 2a_0 <= 18 since a_0 is a digit, so is no greater than 9. On the other hand, we assumed that a_k was not 0, and it is a digit as well, so it is between 1 and 9. Thus, (10^k-3)a_k >= (10^k-3), so that (10^k-3) <= 18. The only way that this can happen is if k=1 or k=0, so that the number has 1 or 2 digits.
If k=1, then the equality giving the property of the number we are interested in is a_1*10 + a_0 = 3a_1 + 3a_0, so that 7a_1=2a_0. Since both sides are integers, unique factorization shows that 7 divides a_0, and since a_0 is a single digit, it must be 7. Then the only possibility for a_1 is that it is 2, so the number is 27.
If k=0, then the equality reads a_0 = 3a_0, which is impossible unless a_0=0, but we assumed the leading digit of our number was nonzero.
2007-01-09 08:55:35 · answer #2 · answered by Anonymous · 1⤊ 1⤋
It has 2 digits, and the number is
10X+y, than:
according to condition
10x + y = 3(x+y)
7x = 2y
x : y => 2 : 7
so,
x = 2 and y = 7
Check the above number
2+7 = 9 = The sum of its digits
9 * 3 = 27
2007-01-10 04:29:00 · answer #3 · answered by Kinu Sharma 2 · 0⤊ 0⤋
27
2007-01-10 05:51:51 · answer #4 · answered by WhItE_HoLe 3 · 0⤊ 0⤋
27
2007-01-09 08:56:18 · answer #5 · answered by S.S.KUMAR 3 · 0⤊ 0⤋
27
2007-01-09 08:47:09 · answer #6 · answered by chaching 2 · 0⤊ 1⤋
27 is the special number...
2 and 7 are its digits.
2+7 = 9 = the sum of its digits
9 * 3 = 27
2007-01-09 08:48:34 · answer #7 · answered by Autisteek 2 · 0⤊ 1⤋
assuming it has 2 digits, and the number is 10X+y, than:
10x + y = 3(x+y)
7x=2y
x=2, y=7
2007-01-09 09:06:18 · answer #8 · answered by Anonymous · 0⤊ 0⤋
this *peculiar* number is 27.
2007-01-09 10:45:47 · answer #9 · answered by Heady 3 · 0⤊ 0⤋
27
EDIT: beaten :(
2007-01-09 08:45:18 · answer #10 · answered by heidavey 5 · 0⤊ 1⤋