With the three digits of the integer 142, there are sixtwo-digit positive intege

Question

With the three digits of the integer 142, there are sixtwo-digit positive integers that can be formed. When the sumof those two-digit integers is divided by the sum of the threedigits of the original three-digit number, the result is22.   When the same thing is done with the k digits of ak-digit integer and each of the k digits is different and non-zero,the result of dividing the sum of all of the possible two-digitintegers by the sum of the k digits of the original k-digit integeris 66. What is the value of k?

Explanation / Answer

The answer is k = 7 and here is why: when we had a 3 digit number and divided the sum of the 2digit numbers by the sum of the 3 digits of the originalnumber it equaled 22. when you do the same experiment with 4 digits w, x, y, and zyou get 33: wx, wy, wz, xw, xy, xz, yw, yx, yz, zw, zx, zy are the twodigit numbers so the sum is (10w +x) + (10w +y) + (10w +z)+ (10x +w) + (10x +y) + (10x +z)+ (10y +w) + (10y +x) + (10y +z)+ (10z +w) + (10z +x) + (10z +y)= 30w + 30x + 30y + 30z + + 3w + 3x +3y + 3 z =33w + 33x + 33y + 33z sum of the 2-digitnumbers
or 33(w + x + y + z) when this is divided by (w + x + y + z)you get 33. In conclusion: 3 digit number gives you 22 4 digit number gives you 33 5 digit number gives you 44 6 digit number gives you 55 7 digit number gives you 66 so K=7

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