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1.One digit of 970,405 can be changed to make the result divisible by 225. What is the six-digit number after the modification?

2. For each positive integer n, let S(n) denote the sum of the digits of n. How many three-digit n's are there such that n+S(n)+S(S(n)) 0 ≡ mod{9}?

3. A school bought many boxes of $100$ candies each. The teachers opened all the boxes and gave 14 candies to each student, with 2 candies leftover at the end. The receipt shows the school bought 10* boxed boxes of candies where the units digit, represented by *, was covered by a chocolate smudge. How many boxes of candies did the school buy?

 Oct 3, 2020
 #1
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1 - Change 970,405 to 970,425 and you have a multiple of 225.

 Oct 3, 2020
 #2
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3 - The school bought 106 boxes of candies, because:

 

[106 * 100] mod 14 ==2

 Oct 3, 2020
 #3
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2 - There are 576 three-digit numbers. Their sum: 5 + 7 + 6 =18. The sum of 18 = 1 + 8 =9. So:

576 + 18 + 9 =603 mod 9 =0.

 Oct 3, 2020

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