Unit Digit

Number Theory Level 1

What is the unit digit of the following product: 684 × 59 × 413 × 76 ? 684 \times 59 \times 413 \times 76?

8 6 4 2

Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Zee Ell
Aug 19, 2016

The quickest way to solve this question (without a calculator) is to multiply the last digits only (as digits of higher place values, e.g. Tens or Hundreds have no effect on the Units digit when multiplying integers) and take a step-by-step approach (carrying over the last digit of the result only).

The last digit of 68 4 × 5 9 × 41 3 × 7 6 : \text {The last digit of } 68 {\color{#3D99F6} 4} × 5{\color{#3D99F6} 9} × 41{\color{#3D99F6} 3} × 7{\color{#3D99F6} 6} \text { : }

4 × 9 = 3 6 {\color{#3D99F6} 4} × {\color{#3D99F6} 9} = 3{\color{#D61F06} 6}

6 × 3 = 1 8 {\color{#D61F06} 6} × {\color{#3D99F6} 3} = 1{\color{#EC7300} 8}

8 × 6 = 4 8 {\color{#EC7300} 8} × {\color{#3D99F6} 6} = 4{\color{#20A900} 8}

Therefore, our answer should be 8 . \text {Therefore, our answer should be } \boxed {8} .

15 Helpful 0 Interesting 0 Brilliant 0 Confused

Is there any way of using the number of multiples of two in the prime factorization of solution to find this result? That was my thought after seeing the answer, and that also wouldn't require a calculator since you would only need to divide each factor by 2 until you no longer can do so.

Cara Wegner - 4 years, 9 months ago

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If you believe, that your method works and it isn't too long, please feel free to post it here as an alternative solution.

Zee Ell - 4 years, 9 months ago

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