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Find a 4-digit number divisible by 9 and 25.

Answer is 9000.

I can find the greatest number which is divisible by 9 and 25 but 9900 != 9000. What am I doing wrong?

9, 25 | 3
3, 25 | 3
1, 25 | 5
1, 5  | 5
1, 1

LCM(9, 25) = 3 * 3 * 5 * 5 = 225
Dividing 9999 by 225, the remainder is 99.
Required number = (9999 - 99) = 9000


The greatest number of four digit which is exactly divisible by 9, 25 is 9900.
nop
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    $25\times 9=225$. Now just multiply by 5, say, to get $1125$ or 40, say, to get $9000$. – pshmath0 Oct 25 '22 at 19:37
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    There are exactly $40$ numbers with $4$ digits and no leading zeros that are divisible by $9$ and $25$. Observe that $\operatorname{lcm}(9,25)=225,\left\lceil\frac{1000}{225}\right\rceil=5,\left\lfloor\frac{9999}{225}\right\rfloor=44$, so the $40$ numbers $225n$ with $n=5,6,\ldots,44$ comprise all such numbers. So "Answer is $9000$" is very incomplete. – Jam Oct 25 '22 at 19:40
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    @pathfinder, looks like what I did is an answer too. If I want to get all the rest of the numbers I just do 9900 - 225 = 9675 and so on and so forth. 1125 | 1350 | 1575 | 1800 | 2025 | 2250 | 2475 | 2700 | 2925 | 3150 | 3375 | 3600 | 3825 | 4050 | 4275 | 4500 | 4725 | 4950 | 5175 | 5400 | 5625 | 5850 | 6075 | 6300 | 6525 | 6750 | 6975 | 7200 | 7425 | 7650 | 7875 | 8100 | 8325 | 8550 | 8775 | 9000 | 9225 | 9450 | 9675 | 9900 – nop Oct 25 '22 at 20:08
  • @Jam, how come 1000 / 225 = 5? Isn't it supposed to be 4 since 9999 / 225 = 44? – nop Oct 25 '22 at 20:08
  • What is the exact textbook question? Those are wrong sometimes! And 9900 obviously meets your criterion. – Pieter21 Oct 25 '22 at 20:09
  • @Pieter21. That was the exact question in the book. It's probably a typo and they wanted to say 9990. – nop Oct 25 '22 at 20:10
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    So it is 'an' answer and not 'the' answer. Case closed.. – Pieter21 Oct 25 '22 at 20:12
  • @nop The brackets I've used there denote the ceiling (round up) and floor (round down) functions respectively. We have that the smallest $4$-digit number has a quotient $1000/225=4.4$, so we round that up to $5$ in order to find the smallest integer multiple of $225$ in the range. – Jam Oct 25 '22 at 20:13
  • @Jam, oh, thank you! – nop Oct 25 '22 at 20:15
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    If you want all of them: the lowest is $1125$ from my first comment. The highest is $225\times 44$. So all of them are given by $225n$ where $n=5,6,7,\ldots,44$. – pshmath0 Oct 25 '22 at 20:22

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