Anthony got a new iPhone. He needs to create a 4-digit passcode. How many passcodes can be created if REPETITION OF DIGITS IS AL
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4, 5, and 7 are mutually coprime, so you can use the Chinese remainder theorem right away.
We construct a number such that taking it mod 4, 5, and 7 leaves the desired remainders:
- Taken mod 4, the last two terms vanish and we have
so we multiply the first term by 3.
- Taken mod 5, the first and last terms vanish and we have
so we multiply the second term by 2.
- Taken mod 7, the first two terms vanish and we have
so we multiply the last term by 7.
Now,
By the CRT, the system of congruences has a general solution
or all integers ,
, the least (and positive) of which is 27.