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Fields Medal Prize Winners (1998)

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Multiplying by 14443

Multiplication by 14443 using 101101

My first instinct when tackling the number 14443 was to figure out why the number 14443 was so special. The first observation was that when 14443 is multiplied by a multiple of 7, the answer is always in the form of n0nn0n, under certain circumstances.

In general, the problem 14443 × t = n0nn0n, where t = 7n , the multiple of 7, and 7 t 69. We require these bounds to be put on t for the product to be a six-digit number. 17

Example:

14443 × 42 =

Since 42 = 7(6) and thus a multiple of 7, we can simply write the product as 606606.

14443 × 42 = 606606.

 

That's a great little trick and it will work about of the time! We needed something a little more powerful and a trick that could be used on the other numbers. The thing to notice about the number 14443 is that it is equal to . Remembering how to multiply by 25 led me to nd this trick. When multiplying by 25, you multiply by 100 and divide by 4 because . Why not try that here? The trick to multiplying by 14443 is to multiply the two-digit number by 101101 first and divide by 7.

What does a two-digit number, when multiplied by 101101, look like? It's pretty easy to visualize actually. Think of the number in three different parts. When you multiply 14443 × N and N is a two-digit number, the three parts are

[ N ] [11 N ] [ N ]

 

Example:

14443 × 34 =

To start 14443 × 34, you think 34 374 34, because 34 × 101101 = 3437434.

The next step is to divide by 7, which you should be able to do. Be careful and watch the carries.

 

Multiplication by 14443 using 13

Further investigation of the number 14443 reveals that it is composite and has a prime factorization of 11 × 13 × 101. Multiplication by 11 and multiplication by 101 are two of the cornerstone tricks for Number Sense. Multiplication by 13 is a little more difficult, but managable. To simplify things even more, the product of 11 and 101 is 1111, which is another easy trick to use.

To multiply by 14443, use the following steps:

  • Multiply the other number by 13. Keep the product in your head.
  • Multiply that product by 1111, and write the answer right to left as you go.

Example:

14443 × 32 =

  • Multiply 32 × 13 = 416. I would use LOIF (FOIL, your choice) to find this first product.
  • Multiply 416 × 1111. Multiplication by 1111 is simply adding the digits (and carries) together right to left, and sweeping through four numbers at a time (there for 4 ones, so sweep 4 numbers).
6 6
6 + 1 = 7 76
6 + 1 + 4 = 11, write 1, carry 1 176
6 + 1 + 4 = 11 + 1 = 12, write 2, carry 1 2176
1 + 4 = 5 + 1 = 6, write 6, no carry 62176
4 462176

Therefore, 14443 × 32 = 462176.

 

Multiplication by 14443 using quotients and remainders

For the problem 14443 × n , use the following steps: [Work from right to left, step to step.]

  • Find q and r where n × 7 = q with a remainder of r .
  • Starting at the right, compute 43 r + q and write down the last two digits of this number. Carry if needed.
  • Next, calculate 4 r + q . Add the carry from the previous product. Write down the last digit and carry if needed.
  • Calculate 14 r + q . Add the previous carry. Write down two digits.
  • Write down q .

Example:

14443 × 31 =

31 ÷ 7 = 4 with a remainder of 3. Thus, q = 4 and r = 3.  
43 r + q = 43(3) + 4 = 133. Write 33, carry 1. 33
4 r + q = 4(3) + 4 = 16 + 1 = 17. Write 7, carry 1. 733
14 r + q = 14(3) + 4 = 46 + 1 = 47. Write 47. 47733
Write 4. 447733

Thus, 14443 × 31 = 447733.