*The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.***G. H. Hardy**

Greetings to all visitors of **Math1089 – Mathematics for All**.

I’m thrilled you’re here. I wanted to express my gratitude for your valuable time spent on **Math1089**. Your visit means a lot to me and I truly appreciate you taking the time from your hectic schedule to check out this blog.

Let’s delve into the beauty of the number **588235294117647**! This lovely number can be found in the fraction **1/17**. We’ll begin by exploring Midy’s theorem and uncovering other intriguing characteristics of this number.

**Midy’s Theorem** If the period of a repeating decimal for *a*/*p*, where *p* is prime and *a*/*p* is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s. In the case of **1/17** (*one*–*seventeenth*), the period is 16. In fact, we have

When we multiply **588235294117647** by other numbers, notice how it yields numbers in the same order but with a different starting point. In fact, the **first digit of every product goes to the end of the number to form the next product**. Otherwise, **the order of the digits is intact**. Moreover, the sum of the digits in each of these product numbers is equal to **72**.

The product of the number 0588235294117647 and 17 generates the following result:

**0588235294117647 × 17 = 9999999999999999**

The product of the number 0588235294117647 and multiples of 17 generates the following set of results:

**A few noteworthy observations are as follows:**

**(a)** With an **increasing multiplier** of 0588235294117647, the **leftmost digits** of the product will be in **ascending order**, such as 1, 2, 3, and so on.

**(b)** As the **multiplier** of 0588235294117647 **increases**, the **rightmost digits **of the product will be in **decreasing order**, such as 8, 7, 6, and so on.

**(c)** When the **sum of the multipliers** is held constant (**equal to 187**), the products are **reverse **of each other.

0588235294117647 × 34 = 19999999999999998

0588235294117647 × 153 = 89999999999999991

0588235294117647 × 51 = 29999999999999997

0588235294117647 × 136 = 79999999999999992

0588235294117647 × 68 = 39999999999999996

0588235294117647 × 119 = 69999999999999993

0588235294117647 × 85 = 49999999999999995

0588235294117647 × 102 = 59999999999999994

**The sum of the digits of the number**

We can write the sum of the digits of the number 0588235294117647 in different ways to get different results. Here are a few examples:

**(a)** The sum of all the digits is

0 + 5 + 8 + 8 + 2 + 3 + 5 + 2 + 9 + 4 + 1 + 1 + 7 + 6 + 4 + 7

= 72

≡ 0 (mod 9).

In other words, it is divisible by 9.

**(b)** The sum of the digits taken two at a time is

05 + 88 + 23 + 52 + 94 + 11 + 76 + 47

= 396

≡ 0 (mod 99).

In other words, it is divisible by 99.

**(c)** The sum of the digits taken four at a time is

0588 + 2352 + 9411 + 7647

= 19998

≡ 0 (mod 9999).

In other words, it is divisible by 9999.

**(d)** The sum of all the digits taken eight at a time is

05882352 + 94117647

= 99999999

≡ 0 (mod 99999999).

In other words, it is divisible by 99999999.

We can show that 16 is the smallest value of *n* such that 10* ^{n}* – 1 is divisible by 17. In that case, (10

^{16}– 1) / 17 =

**0588235294117647**(16 digits) and we get

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