If the number of distinct five-digits $1A54B$ is divided by $24$ , then the remainder is 15, what is the sum of the different values ​A can take?

Question

If the number of distinct five-digits $1A54B$ is divided by $24$ , then the remainder is $15$, what is the sum of the different values ​​A can take?

a) 6

b) 8

c) 10

d) 12 .

e) 15

Answer 1

$1A54B =10000 + 1000*A + 540 + B=$

$(24*416 +16) + (24*41 + 16)A+ (22*24+ 12)+B=$

$24(416+41*A+22) + 16*A + 28 + B = $

$24(416+41*A+23) + 16*A + 4 + B$.

So there is some $m$ where $16*A + 4 + B = 24m + 15$

or $16*A = 24m + 11 - B$

$8|16*A$ and $8|24m$ so $8|11-B$ but the only multiple of $8$ between $11 - 9=2$ and $11 + 0=11$ is $8$. so $11 -B = 8$ and $B=3$. So $16*A = 24m + 8$ and $2*A = 3m + 1$.

Okay..... so if $2*A = 0,2,4,6,8,10,12,14,16,18$ and the ones that can be $3m+1$ are $2*A = 4, 10,16$ so $A = 2,5, 8$

But the problem said the digits are distinct so $A \ne 1,5,4,$ or $3$ so $A\ne 5$.

So $A = 2$ or $8$.

And the sum is $10$.

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That was the real hard way.

But if when you divide by $24$ you have remainder $15$ means there is an $\omega$ so that $1A54B = 24\omega + 15= 3(8\omega + 5)$ so $1A54B$ is divisible by $3$. And $24\omega + 15 = 8(3\omega + 1) + 7$ so when you divide $1A54B$ you get remainder of $7$.

$1000 = 8*125$ so $1A000$ is divisible by $8$ so $54B$ will have remainder $7$ when divided by $8$ and $54B = 540 + B$. $540= 8*67+4$ so $4+B=7$. So $B = 3$.

So $1A54B = 1A543$. And this is divisible by $3$ so the sum of the digits are a multiple of $3$. $1+A+5+4+3 = 13+A$. For the sum to be a multiple of $3$, $A$ must equal $2,5,$ or $8$.

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For what its worth: $12543 = 24*522+15$ and $18543 =24*722 + 15$.

Answer 2

Let $N=1A54B +1$. We know that $N\equiv16\pmod {24}$, so $N\equiv0\pmod8$.

1000 is divisible by 8, so $54B+1$ must be divisible by 8. A quick calculator check ensures that $B=3$.

Since 24 and 15 are both multiples of 3, $1A543$ is divisble by 3. This means that the sum of its digits is divisble by 3. Therefore, $A$ must be 2, 5, or 8. But the problem states that the digits must be distinct, so 5 is not allowed. Therefor the answer is 10.