Question

The digits 1,2,3,4,5, and 6 are each used once to compose a six digit number abcdef, such that the three digits number abc is divisible by 5, cbe is divisible by 3, and def is divisible by 11. The digit a is?

Answer 1

!!

Answer 2

a = 2.

Answer 3

Does
e = 4
b=6
c=5
d=3
f=1

Answer 4

Satisfy, right?

Answer 5

how did you get that?

Answer 6

the numbers, that satisfies these conditions, are:
645132
265143
645231
315264
265341
315462

Answer 7

Okay.
Condition one : abc divisible by 5.
Implies c = 5
Now 5 + b + e must be a multiple of 3.
Now make cases. Which is quite easy here.
Max multiple of three possible = 6 + 5+4 = 15
Therefore cases are.
5 b e
4 6 / 6 4 ( When total is 15, b + e = 10)
6 1 / 1 6 ( total = 12 , b + e =7)
5 2 / 2 5 ( ")
4 3 / 3 4 (") ---- Less than that - Not three digit.

Answer 8

Also e = 5 -- Not possible ( c is 5 - One case eliminated)

Now condition 3,
d + f - e = Multiple of 11.
Has to be 0, As 11-- Not possible.
Therefore d + f = e.
Possible values of e = 6,4,1,2,
1 , 2 --- Not possible as sum of two numbers from given list.

Answer 9

And 3^

Answer 10

Yes. That gives all the cases mathmagician drew up :)

Answer 11

thank!