A number is divisible by if and only if the last digits of the number are divisible by . Thus, in particular, a number is divisible by 2 if and only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8.
A number is divisible by 3 or 9 if and only if the sum of its digits is divisible by 3 or 9, respectively. Note that this does not work for higher powers of 3. For instance, the sum of the digits of 1899 is divisible by 27, but 1899 is not itself divisible by 27.
Rule 3: "Tail-End divisibility." Note. This only tells you if it is divisible and NOT the remainder. Take a number say 12345. Look at the last digit and add or subtract a multiple of 7 to make it zero. In this case we get 12380 or 12310 (both are acceptable; I am using the former). Lop off the ending 0's and repeat. 1238 - 28 ==> 1210 ==> 121 - 21 ==> 100 ==> 1 NOPE. Works in general with numbers that are relatively prime to the base (and works GREAT in binary). Here's one that works. 12348 - 28 ==> 12320 ==> 1232 +28 ==> 1260 ==> 126 + 14 ==> 14 YAY!
Divisibility Rule for 10 and Powers of 10
If a number is power of 10, define it as a power of 10. The exponent is the number of zeros that should be at the end of a number for it to be divisible by that power of 10.
Example: A number needs to have 6 zeroes at the end of it to be divisible by 1,000,000 because .
Divisibility Rule for 11
A number is divisible by 11 if and only if the alternating sum of the digits is divisible by 11.
Rule 1: Truncate the last digit, multiply it by 4 and add it to the rest of the number. The result is divisible by 13 if and only if the original number was divisble by 13. This process can be repeated for large numbers, as with the second divisibility rule for 7.
For every prime number other than 2 and 5, there exists a rule similar to rule 2 for divisibility by 7. For a general prime , there exists some number such that an integer is divisible by if and only if truncating the last digit, multiplying it by and subtracting it from the remaining number gives us a result divisible by . Divisibility rule 2 for 7 says that for , . The divisibility rule for 11 is equivalent to choosing . The divisibility rule for 3 is equivalent to choosing . These rules can also be found under the appropriate conditions in number bases other than 10. Also note that these rules exist in two forms: if is replaced by then subtraction may be replaced with addition. We see one instance of this in the divisibility rule for 13: we could multiply by 9 and subtract rather than multiplying by 4 and adding.
More general note for composites
A number is divisible by , where the prime factorization of is , if the number is divisible by each of .
Is 55682168544 divisible by 36?
First, we find the prime factorization of 36 to be . Thus we must check for divisibility by 4 and 9 to see if it's divisible by 36.
Since the last two digits, 44, of the number is divisible by 4, so is the entire number.
To check for divisibility by 9, we look to see if the sum of the digits is divisible by 9. The sum of the digits is 54 which is divisible by 9.
Thus, the number is divisible by both 4 and 9 and must be divisible by 36.
Measuring Time Logic Puzzle
You are given with two ropes with variable width. However if we start burning both the ropes, they will burn at exactly same time i.e. an hour. The ropes are non-homogeneous in nature. You are asked to measure 45 minutes by using these two ropes.
How can you do it?
Please note that you can’t break the rope in half as it is being clearly stated that the ropes are non-homogeneous in nature. Answer & ExplanationSolution:
All you have to do is burn the first rope from both the ends and the second rope from one end only simultaneously. The first rope will burn in 30 minutes (half of an hour since we burned from both sides) while the other rope would have burnt half. At this moment, light the second rope from the other end as well. Where, the second rope would have taken half an hour more to burn completely, it will take just 15 minutes as we have lit it from the other end too.
Thus you have successfully calculated 30+15 = 45 minutes …
Select 01-Select All Given a City table, whose fields are described
as +-------------+----------+ |
Field | Type | +-------------+----------+ |
ID | int(11) | |
Name | char(35) | |
CountryCode | char(3) | |
District | char(20) | |
Population | int(11) | +-------------+----------+ write a query that will fetch all columns for every row in the
table. My Solution SELECT*FROM city; --------------------------------------------------------------------------------- 02-Select by ID Given a City table, whose fields are described