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30MSC 2023 #5: Number Splitting.

30MSC 2023 #5: Number Splitting.

Many arithmetic problems can be solved mentally by number splitting. It is done partitioning a long number into two or more groups by their decimal places. The shorter parts, then become easy to process and compute.

Depending on the need, 1234 can be split into several ways, including the following:

1 | 2 | 3 | 4 :  1,000 + 200 + 30 + 4

12 | 34 : 1200 + 34

1 | 23 | 4 : 1,000 + 230 + 4

This technique is handy in solving this problem given in the Intermediate group of the 1st Math2Shine International Vedic Mathematics Competitions held last July, 2023:

Which number is not divisible by 7.

  1. 1428735;  b) 2874935; c) 4284721; d) 7144263;  e) 5617084

But before we determine the answer, let us first be reminded that attaching any number of zeroes to the end of a number does not affect its divisibility by a divisor. For example, 26 is divisible by 13, so 260 or 2,600 is divisible by 13.

It follows that we can remove some extra zeroes from the end and still maintain its divisibility by that divisor. 200,000 is divisible by 8. We can remove 1, 2 or 3 zeroes from the end but not 4 to maintain its divisibility by 8.

Now let us partition each choice into digit groups divisible by 7.

  1. 1428735  → 1400000 + 28000 + 700 + 35 → 14 | 28 | 7 | 35 – Divisible  
  2. 2874935 → 28 | 7 | 49 | 35 – Divisible
  3. 4284721 → 42 | 84 | 7 | 21 – Divisible
  4. 7144 263 → 7 | 14 | 42 | 63 – Divisible
  5. 5617084 → 56 | 1 | 70 | 84  – not Divisible

Choice e can be decomposed as 5,600,000 + 10,000 + 7,000 + 84. We can see that 5,600,000 , 7,000 and 84 are all divisible by 7 but 10,000 is not.

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