2007-02-09 05:06:02 · 7 answers · asked by keshu 1 in Science & Mathematics ➔ Mathematics
emm,
i think the question is:
if (a+11b)is divisible by 13 and (b+11a)is divisible by 13. a,b are positive integers.Then what is least (a+b) ?
this is a nice,and simple enough problem:
so:
13 | a + 11b and 13 | 11a + b
so: 13 | a + 11b + 11a + b => 13 | 12a + 12b => 13 | 12(a + b)
and it's obvious that: 13 | 13(a + b)
so we get: 13 | 13(a+b) - 12(a+b) => 13 | (a + b)
it means that (a + b) = 13k
good job!
now we are to find the least value of (a+b):
a + b = 13k => b = 13k - a
putting it in the condition:
13 | 11a + b => 13 | 11a + (13k - a) => 13 | 10a + 13k
we know that 13 | 13k
then we can conclude that: 13 | 10a
mmm,a can't be zero,so the least value for a is 13 so that:
13 | 10(13) = 130
so,min (a) = 13 ,then:
13 | a + 11b => 13 | 13 + 11b => 13 | 11b
and again,it mean's that the least value for b,is 13.
we are there,the least value for (a+b) is 26
and it happens when: a = b = 13
good luck
2007-02-09 06:11:52 · answer #1 · answered by farbod f 2 · 0⤊ 0⤋
3, a=2 b=1
2007-02-09 13:14:14 · answer #2 · answered by matrix man 2 · 0⤊ 0⤋
3, b=1 a=2
2007-02-09 13:10:39 · answer #3 · answered by san 3 · 0⤊ 0⤋
(a+11b) / 13 = 0
--> a= -11b
--> a + 11 * b
2 + 11 * 1
4 + 11 * 2
. + 11 * .
smallest value of a+b would be 3 (a=2,b=1)
2007-02-09 13:25:42 · answer #4 · answered by hbk 1 · 0⤊ 0⤋
The least value of a+ b is 3. a = 2, b = 1.
2007-02-09 13:16:40 · answer #5 · answered by steiner1745 7 · 0⤊ 0⤋
Ans: a=2 b=1
2007-02-10 01:50:15 · answer #6 · answered by chaitali 1 · 0⤊ 0⤋
your question is from RMO 2006 which i also gave. the question is bit different. thea answer was 28
2007-02-10 07:33:44 · answer #7 · answered by Apoorv g 2 · 0⤊ 0⤋