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Visualizing the Numbers (Part VI)

© Vidya Narayan Wadadekar

In the last article we had seen the outcome of dividing an even number with an even number. In this article we will try to summarize our observations for the remaining combinations of division.

In the figure to your left-hand side results of the division of an even number with an odd number are summarized. In this case if the quotient is an even number we will be subtracting an even number from an even number. Obviously, the remainder will be either an even number or zero. However, if the quotient is an odd number we will be subtracting an odd number from an even number. Hence, the remainder will be an odd number.

Observe this in the examples below:

• 6 divided by 3 gives 2 with remainder 0.
• 8 divided by 3 gives 2 with remainder 2.

In these examples the quotient is an even number. In the first case, we have zero remainder and in the second case it is an even number.

Now observe the example given below:

• 10 divided by 3 gives 3 with remainder 1.

The summary of the result of the division of an odd number with an even number is shown in the figure to the right-hand side. In this case whether quotient is even or odd we subtract an even number from an odd number. Therefore, there is always some remainder and it is always an odd number. Examples are given below:

• 9 divided by 4 gives 2 with remainder 1.
• 15 divided by 4 gives 3 with remainder 3.

In the first example the quotient is an even number. Whereas, in the second example the quotient is an odd number. In both the cases the remainder is an odd number.

The last observation summarized in the figure to your left-hand side is for the division of an odd number with an odd number. As in the case of division of an even number with an odd number, in this case also whether the remainder is an odd or even number depends on the quotient. However, if the quotient is an even number the remainder is an odd number. And if the quotient is an odd number the remainder is either zero or an even number. Let us see some examples here:

• 13 divided by 3 gives 4 with remainder 1.

In this case the quotient being an even number we are subtracting an even number from an odd number. The obvious result of this is the odd remainder.

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