Last Digits

Number Theory Level pending

What are the last 8 digits of : 11 × 101 × 1001 × 10001 × 100001 × 1000001 × 111 ? 11 \times 101 \times 1001 \times 10001 \times 100001 \times 1000001 \times 111 ?


Source: MIT- Harvard Math Tournament.
12345678 19765438 19876542 19754321

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2 solutions

Ayush G Rai
Nov 5, 2016

Well if you carefully observe,the last digit should be 1 and in the options only 19754321 \boxed{19754321} matches.

Omg, I can't edit the answers now. Thanks for the nice observation.

Hana Wehbi - 4 years, 7 months ago

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Haha.lol.Your most welcome.You can ask @Calvin Lin to edit the answers.

Ayush G Rai - 4 years, 7 months ago

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I will. Gotta make it little bit hard :)

Hana Wehbi - 4 years, 7 months ago
Hana Wehbi
Nov 5, 2016

Multiply first 11 × 101 × 10001 = 11 , 111 , 111 11\times 101\times 10001= 11,111,111 ,

then multiply 111 × 1001 × 1000001 = 111 , 111 , 111 , 111 111\times 1001\times 1000001= 111,111,111,111 ,

the last eight digits of 11 , 111 , 111 × 111 , 111 , 111 , 111 11,111,111\times 111,111,111,111 is 87654321 87654321 .

Now, the last 8 8 digits of 87654321 × 100001 = 87654321 + . . . 32100000 87654321 \times 100001 = 87654321+...32100000 which are 19754321 19754321

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