6

Here is what observed for the results of $m\, 9$s $\times n\, 9$s ($m \leq n$)

$9\times9=81, 9\times99=891, 9\times999=8991$

$99\times99=9801, 99\times 999=98901, 99\times9999=989901$

$999\times999=998001,999\times 9999=9989001, 999\times99999=99899001$

Summarize the following trick to get the result fast:

For example to get $999\times99999 =99899001$

  1. Write down $n\,$$9$s $(n=5)$ first : $\to 99999$
  2. Replace the $m\,$th $(m=3) 9\,$s with $8: \to 99899$
  3. Attach $ (m-1)\, 0$s $(m-1=2): \to 99899 00$
  4. Attach last $1: \to 99899 00 1$

My question is how to prove this is true with algebra?

Star Bright
  • 2,338
  • 7
    Hint: the number composed of $m$ $9$s equals $10^m-1$. – Greg Martin Mar 04 '21 at 21:22
  • 2
    Thanks! As simple as that? Figured out the key is to add and substract $10^x$ in between and pay attention to the n's digit from left: $(10^m-1)(10^n-1)$= $10^{m+n}-10^m-10^{n}+1$

    =$(10.10^{m+n-1}-10^{m+n-1})+(10.10^{m+n-2}-10^{m+n-2})+...+[$only 8$ ] (10.10^{m+n-n}-2.10^{m})$+ $ (10.10^{m-1}-10^{m-1})+....$ last 9 $+0.10^{n-1}+...+0.10^1+1$

     – Star Bright Mar 04 '21 at 23:21

2 Answers2

3

Here's a setup to clarify $m=4,n=6$: \begin{align} 9999 \times 999999 &= 999999 \times (10000-1)\\ &=9999990000 - 999999 \\ &= 9999990000 - 1000000 + 1 \\ &= 999\color{#0B3}{8}99000\color{#0B3}{1}\\ \end{align}

So looking at this semi-lexicographically, we shifted the $n$ $9$s by $m$ trailing zeroes, then switched the last zero to a $1$ and the $m^{th}$ $9$ to an $8$. Which is exactly what you found experimentally.

Joffan
  • 39,627
3

This is an answer to a question that includes your question.

Just as you can cast out $9$s, you can also cast out $99$s, $999$s, etc.

For example.

  • $98765 \equiv 9+8+7+6+5 \equiv 35 \equiv 3+5 \equiv 8 \pmod 9$

  • $98765 \equiv 09+87+65 \equiv 161 \equiv 01 + 61 \equiv 62\pmod{99}$

  • $98765 \equiv 098 + 765 \equiv 863 \pmod{999}$

To multiply a single digit number, $n$ times $9$:

  • The first digit is $n-1$ and the second digit is $9 - (n-1)$.
  • For example, $3 \times 9 = 27 \qquad (2+7=9)$
  • For example, $9 \times 9 = 81 \qquad (8+1=9)$

To multiply a two-digit digit number, $ab$ times $99$:

  • The first two digits are $ab-1$ and the second two digits are $99 - (ab-1)$.
  • For example, $32 \times 99 = 3168 \qquad (31+68=99)$
  • For example, $40 \times 99 = 3960 \qquad (39+60=99)$
  • For example, $99 \times 99 = 9801 \qquad (98+01=99)$

To multiply a three-digit digit number, $abc$ times $999$:

  • The first three digits are $abc-1$ and the second three digits is $999 - (abc-1)$.
  • For example, $342 \times 999 = 341658 \qquad (341+658=999)$
  • For example, $44 \times 999 = 043956 \qquad (043+956=999)$
  • For example, $999 \times 999 = 998001 \qquad (998+001=999)$

One more example:

  • $123456 \times 999999 = 123455876544$
Steven Alexis Gregory
  • 26,769