5-digit Number Puzzle

a 474 b \overline{a474b} is a five-digit number that is divisible by 63.

Find the minimum value of a + b a+b .

5 13 3 9

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1 solution

Chew-Seong Cheong
Feb 27, 2017

For a a 474 b b to be divisible by 63 it must be also divisible by 9. When an integer is divisible by 9, the sum of its digits is also divisible by 9. Since 4 + 7 + 4 = 15 4+7+4 = 15 , a + b = a+b = 3 or 12; and the possible solutions are:

{ a + b = 3 { a = 1 , b = 2 63 14742 a solution a = 2 , b = 1 63 24741 not a solution a + b = 12 { a = 3 , b = 9 63 34749 not a solution a = 4 , b = 8 63 44748 not a solution a = 5 , b = 7 63 54747 a solution a = 6 , b = 6 63 64746 not a solution a = 7 , b = 5 63 74745 not a solution a = 8 , b = 4 63 84744 not a solution a = 9 , b = 3 63 94743 not a solution \implies \begin{cases} a+b=3 & \implies \begin{cases} a=1, b=2 & \color{#3D99F6} 63 \ | \ 14742 \small \text{ a solution} \\ a=2, b=1 & \color{#D61F06} 63 \nmid 24741 \small \text{ not a solution} \end{cases} \\ a+b = 12 & \implies \begin{cases} a=3, b=9 & \color{#D61F06} 63 \nmid 34749 \small \text{ not a solution} \\ a=4, b=8 & \color{#D61F06} 63 \nmid 44748 \small \text{ not a solution} \\ a=5, b=7 & \color{#3D99F6} 63 \ | \ 54747 \small \text{ a solution} \\ a=6, b=6 & \color{#D61F06} 63 \nmid 64746 \small \text{ not a solution} \\ a=7, b=5 & \color{#D61F06} 63 \nmid 74745 \small \text{ not a solution} \\ a=8, b=4 & \color{#D61F06} 63 \nmid 84744 \small \text{ not a solution} \\ a=9, b=3 & \color{#D61F06} 63 \nmid 94743 \small \text{ not a solution} \end{cases} \end{cases}

We note that the only acceptable option is 3 \boxed{3} .

While 12 isn't a listed option, 54747 is also divisible by 63, so both 3 and 12 are acceptable solutions.

Steve McMath - 4 years, 3 months ago

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Thanks. I see that this problem has been edited.

In future, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the menu. This will notify the problem creator (and eventually staff) who can fix the issues.

Brilliant Mathematics Staff - 4 years, 3 months ago

Thanks. I was just that it is not in the options.

Chew-Seong Cheong - 4 years, 3 months ago

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