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Last 2 digits of a Long Multiplication

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Last 2 digits of a Long Multiplication

shape Introduction

How to find the last 2 digits of a long multiplication?

In this article, we will be studying how to calculate the last 2 digits of a long multiplication. We will understand the methodology using some examples.


shape Examples

1. Find the last two digits of 15 x 37 x 63 x 51 x 97 x 17.

Solution:

    Step 1: Divide the expression by 100. \(\frac{15 \times 37 \times 63 \times 51 \times 97 \times 17}{100}\).


    Step 2: Reduce this expression to remove all possible factors from numerator and denominator.

      \(\frac{\not{15}^{3} \times 37 \times 63 \times 51 \times 97 \times 17}{\not{100}^{20}}\) = \(\frac{3 \times 37 \times 63 \times 51 \times 97 \times 17}{20}\)


    Step 3: Since, we can’t reduce further, now check how much is each term less/more than multiple of denominator. e.g.

      – 37 is 3 less than (20 times 2 = 20 × 2 = 40)

      – 63 is 3 more than (20 times 3 = 20 × 3 = 60)

      – 51 is 9 less than (20 times 3 = 20 × 3 = 60)

      \(\frac{(+3) \times (-3) \times (+3) \times (-9) \times (-3) \times (-3)}{20}\) = \(\frac{(27) \times (81)}{20}\)


    Step 4: Apply Step 3 again to this fraction.

      \(\frac{(+7) \times (+1)}{20}\)


    Step 5: Make the denominator 100 again.

      \(\frac{7}{20}\) x \(\frac{5}{5}\) = \(\frac{35}{100}\)

      The numerator obtained gives the answer, i.e. 35.

      Check: 15 × 37 × 63 × 51 × 97 × 17 = 2940521535, which is correct.


    2. Find the last two digits of 65 x 29 x 37 x 63 x 71 x 87 x 62.

    Solution:

    Step 1: Divide the expression by 100. \(\frac{65 \times 29 \times 37 \times 63 \times 71 \times 87 \times 62}{100}\).


    Step 2: Reduce this expression to remove all possible factors from numerator and denom- inator.

      \(\frac{\not{65}^{13} \times 29 \times 37 \times 63 \times 71 \times 87 \times 62}{\not{100}^{20}}\)

      = \(\frac{13 \times 29 \times 37 \times 63 \times 71 \times 87 \times \not{62}^{31}}{\not{20}^{10}}\)

      = \(\frac{13 \times 29 \times 37 \times 63 \times 71 \times 87 \times 31}{10}\)


    Step 3: Since, we can’t reduce further, now check how much is each term less/more than multiple of denominator.

      = \(\frac{(+3) \times (-1) \times (-3) \times (+3) \times (+1) \times (-3) \times (+1)}{10}\)

      = – \(\frac{81}{10}\)


    Step 4: Apply Step 3 again to this fraction. \(\frac{(-1)}{10}\)


    Step 5: Make the denominator 100 again.

    We can’t make the denominator 100 directly because the numerator is negative (last two digits cannot be negative!) Therefore, add the numerator to the denominator. This gives

      \(\frac{(-1 + 10)}{10}\) = \(\frac{9}{10}\)


    Now, since the numerator is positive, we can convert the denominator to 100 and get our answer.

      \(\frac{9}{10}\) x \(\frac{10}{10}\) = \(\frac{90}{100}\)


    The numerator obtained gives the answer, i.e. 90.

      Check: 65 × 29 × 37 × 63 × 71 × 87 × 62 = 1682762862690, which is correct.


shape Exercises

1. Find out the last two digits of the expression 201 × 202 × 203 × 204 × 246 × 247 × 248 × 249.

2. Find out the last two digits of the expression 301 × 402 × 503 × 604.

Answers:


    1. 56

    2. 24