Numbers Ending With…

Question 1

What it the end of a number? (Definition)

Answer

The end of a number is the digit (or list of digits) located on the right in its decimal notation.

For an integer, the last digit is the units digit.

For a finite floating point number it is the last digit of its finite decimal representation.

For a floating point number with non-finite decimal expansion, the number has no end.

Question 2

How to list numbers that ends with a given digit?

Answer

To list numbers with the same end digits, it is possible to browse all the integers in ascending order and check their last digits.

Another faster method is to browse these same integers and to concatenate the final digit(s) to it.

Example: 5, 15, 25, 35 are the first numbers ending with 5 (and therefore all numbers are multiples of 5)

Question 3

How to list multiples of some numbers?

Answer

Some digits positioned at the end of the number make it possible to deduce that the number is a multiple of N (under some conditions).

Example: A number ending in 00 is necessarily multiple of 100 (but also 50, 25, 10, 5, and 2)

Example: A number finishing in 5 is necessarily multiple of 5

Example: A number ending in 0, 2, 4, 6 or 8 is necessarily multiple of 2

Question 4

What is the name of numbers ending with '0'?

Answer

Numbers that finish with one or more 0 are multiples of 10, sometimes called round numbers.

Mathematically speaking, a round number is different, it is an integer that does not have a decimal point nor decimal places, only an integer part.

Question 5

How to determine the parity of a number with its last digit?

Answer

If the last digit of an integer is 0, 2, 4, 6 or 8 then the number is even.

If the last digit of an integer is 1, 3, 5, 7 or 9 then the number is odd.

Question 6

What is an automorphic number?

Answer

An automorphic number is a natural integer $n$ whose squared value $n^2$ ends with the same digits as the number itself.

Example: 76^2 = 5776

Example: 625^2 = 390625

Question 7

How to extract the last digit of a number in programming?

Answer

Several techniques are possible, depending on whether the number is stored as an integer or a string.

If the number is an integer, then the modulo operator (%) can be used.

Example: last_digit = number % 10

If the number is a string, then retrieving the last digit can be done with string operators like substr

Question 8

How to find a phone number ending in XX?

Answer

To find a phone number that ends with specific digits, there are several options:

— Check your own contacts (to search for a partial number among your contacts, use the search bar in your phone's Contacts application and enter the last digits.

— Some platforms like WhatsApp, Telegram or Facebook Messenger allow you to search for contacts by phone number.

— List all the numbers (there can be several thousand) and call them (or search for them on the web).

Numbers Ending With…