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Consider the following problem. All the even numbers from 2 to 128 (both inclusive) excepting the numbers end with 0 are multiplied together. What is the unit’s digit of the product?
Teacher’s tool: If a problem is too big then consider a simple problem with the same conditions.
Solution: Consider the numbers 2, 4, 6, 8.
These are the numbers from 2 to 8 (both inclusive) and even, where is 0 is not there in the unit place.
The product is 2 × 4 × 6 × 8 = 384. The unit’s digit is 4.
Consider the even numbers between 2 and 18 (both inclusive and unit digit non zero) 2, 4, 6, 8, 12, 14, 16, 18.
We observe that the unit’s digit by multiplying 2, 4, 6, 8 is 4. Clearly the unit’s digit of multiplying the numbers 12, 14, 16 and 18 also must be 4.
∴ The unit’s digit in the product must be 6.
Take numbers between 2 and 28 with the same conditions.
(2, 4, 6, 8) (12, 14, 16, 18) (22, 24, 26, 28). The units digit in the final product must be 4.
So, there is a pattern formed.
Table:
Group | 1 | (1, 2) | (1, 2, 3) | (1, 2, 3, 4) | (1, 2, 3, 4, 5) |
Unit digit in the product | 4 | 6 | 4 | 6 | 6 |
Now the given question can be solved easily.
The unit’s digit in the product must be 4 because it is the 11th group.
Note for the teachers: For a gifted child of higher grade a good question will be when all the even numbers between 2 and 198 (both inclusive-deleting the numbers with O in the unit digit) are multiplied, what is the tens digit in the product?
Of course, this is not as simple as the first one. But a teacher can follow the same advice as to consider simple problem along the same line.
What we find here is that the last two digits of the products of the groups is 84. When we multiply all the numbers in two groups, because we need the ten digit of the product only, a simple calculation gives
Thus the tens digit repeats at the 11th stage. The tens digit of the product 2 × 4 × 6 × 8 × 12 × 14 × 16 × 18 is 5.
(i.e.) if we take (2, 4, 6, 8) as the first group and
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(12, 14, 16, 18), as the second group
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(22, 24, 26, 28), as the third group
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(92, 94, 96, 98) is the 10th group.
Let I denote the first group, II the first and second group etc.
Group | I | II | III | IV | V | VI | VII | VIII | IX | X | XI | XII | XIII |
Ten’s digit | 8 | 5 | 0 | 3 | 2 | 1 | 4 | 9 | 6 | 7 | 84 | 5 | 0 |
Now when this analysis is done then the framing of the question by a teacher becomes easy.
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Santhanam, S.R. (2017). Workshop on Framing Non-routine Problems in Mathematics for Gifted Children of Age Group 11–15. In: Kaiser, G. (eds) Proceedings of the 13th International Congress on Mathematical Education. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-62597-3_137
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