Find The Pattern - 2

Number Theory Level 1

We are given the equations below.

987654321 × 9 = 08 888 888 889 987654321 × 18 = 17 777 777 778 987654321 × 27 = 26 666 666 667 987654321 × 36 = 35 555 555 556 987654321 × 45 = 44 444 444 445 987654321 × 54 = 53 333 333 334 987654321 × 63 = 62 222 222 223 \begin{array}{ccrc} 987654321&\times&9&=&08\,888\,888\,889\\ 987654321&\times&18&=&17\,777\,777\,778\\ 987654321&\times&27&=&26\,666\,666\,667\\ 987654321&\times&36&=&35\,555\,555\,556\\ 987654321&\times&45&=&44\,444\,444\,445\\ 987654321&\times&54&=&53\,333\,333\,334\\ 987654321&\times&63&=&62\,222\,222\,223 \\ \end{array}

What is the value of 987654321 × 72 ? 987654321 \times 72 ?

Note : Can you explain what is happening?


The answer is 71111111112.

Chung Kevin by Chung Kevin , 46, Australia

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1 solution

Chew-Seong Cheong
Jul 24, 2016

We note that the pattern is 987654321 × 9 n = a b b b b b b b b b ⏟ 9 b ’s c ‾ 987654321 \times 9n = \overline{a\underbrace{bbbbbbbbb}_{9 \ b \text{'s}}c} , where a = n − 1 a=n-1 , b = 9 − n b=9-n and c = 10 − n c=10-n .

Therefore, for n = 8 n=8 , we have 71111111112 \boxed{71111111112} .

For what is happening:

For positive integer n < 10 n < 10 , the general equation is as follows:

987 654 321 × 9 n = 8 888 888 889 n = 8 888 888 888 n + n = 8 n ( 1 111 111 111 ) + n = ( 9 − 1 ) ( 1 111 111 111 ) n + n = 9 999 999 999 n − 1 111 111 111 n + n = 10 000 000 000 n − n − 1 111 111 111 n + n = 10 000 000 000 n − 1 111 111 111 n = n 0 000 000 000 ‾ − n n n n n n n n n n ‾ = a b b b b b b b b b c ‾ where a = n − 1 , b = 9 − n , c = 10 − n \begin{aligned} 987\ 654\ 321 \times 9n & = 8 \ 888 \ 888 \ 889n \\ & = 8\ 888\ 888\ 888n + n \\ & = 8n(1\ 111\ 111\ 111) + n \\ & = (9-1)(1\ 111\ 111\ 111)n + n \\ & = 9\ 999\ 999\ 999n - 1\ 111\ 111\ 111n + n \\ & = 10\ 000\ 000\ 000n - n - 1\ 111\ 111\ 111n + n \\ & = 10\ 000\ 000\ 000n - 1\ 111\ 111\ 111n \\ & = \overline{n0\ 000\ 000\ 000} - \overline{n\ nnn\ nnn\ nnn} \\ & = \overline{ab\ bbb\ bbb\ bbc } \quad \quad \small \color{#3D99F6}{\text{where }a = n-1, \ b= 9-n, \ c= 10-n} \end{aligned}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Yes, that is the pattern. Now, why is it happening? Note that the pattern breaks when we try and multiply by 90.

Calvin Lin Staff - 4 years, 10 months ago

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Thanks for explaining it :)

The pattern "breaks" when we multiply by 9 × 10 9 \times 10 where n = 10 n = 10 gives us b = − 1 b = - 1 . If we substituted that into the final formula and did the "negative carry over", we will arrive at the value of 88888888890.

Calvin Lin Staff - 4 years, 10 months ago

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