**What is the highest number such that when expressed in letters it contains the five vowels, without repeating them?**

For example in the number “*One billion three*”(1,000,000,000.003), we lack the letter“ a ”

#### Solution

If we start with the largest number with a defined name, the Googol, we see that no number of this magnitude can meet the condition that the vowels are not repeated since it has three letters "o".

Now, if we continue with the "x-illon" and going down we see that it is impossible for any of those "x-illones" to fulfill the condition of non-repeated vowels.

In the "Octillon" it is obvious as it has two letters "o".

The plurals of septillón and sextillón (septillones and sextillones) discard them as they repeat the letter "e". If it were a septillon or a sextillon we would have no way of placing the “a” that we would be missing without repeating another vowel.

The quintillon we discarded as it repeats the letter "i".

In the quadrillion, we only need the “e”, that if it is plural, some vowel is always repeated (eg three quadrillions) and if it is a quadrillion, we cannot add a number that only contributes an “e” and no other vocal.

The trillion cannot be singular because then we would lack the "a" and the "e" and there is no number that only has those two vowels. with the plural of trillions the same thing happens, we would be missing the “a” and the “u” and there is no number that contributes only those two vowels.

The same happens with the trillion and the million as with the trillion.

Finally we reach the thousands. The number cannot contain “-hundred thousand” or be “one hundred thousand” since the “i” is repeated so we have to look for a number from 1 to 99 that contributes “a”, “e”, ”or”, "U" to a thousand. The "nineties" have "a", "e", "or", so only one "u" would be missing. And that "u" is achieved with "a". so we get the **number ninety one thousand**: 91000 as the largest number containing all the vowels once each.