x ≡ 0 mod 2
x ≡ 1 mod 3
x ≡ 2 mod 5
x ≡ 3 mod 7
a. What is the smallest x?
b. What is the largest 4-digit x?
a.
x is even, 2 + 2k
x is 1 mod 3, so 4 + 6k
x is 2 mod 5, so 22 + 30k
x is 3 mod 7:
22 is 1 mod 7
30 is 2 mod 7
answer is 52
b.
10000 is
0 mod 2
1 mod 3
0 mod 5
4 mod 7
we need to subtract y
y is even, 2 + 2k
y is 0 mod 3, so 6 + 6k
y is 3 mod 5, so 18 + 30k
y is 1 mod 7:
18 is 4 mod 7
30 is 2 mod 7 so k is 2
answer is 10000 - (18 + 30*2)
= 10000 - 78 = 9922
Note: in each of these, of course, successive k's are not the same, but merely indicate separation of possible successive values.
Edited on July 2, 2015, 9:22 am
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Posted by Charlie
on 2015-07-02 09:19:53 |
solution with some explanation
