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Well, darling, to find the units digit of 2 to the 57th power, you just need to look for a pattern. The units digit of powers of 2 cycles every 4 powers: 2, 4, 8, 6. So, 57 divided by 4 leaves a remainder of 1, meaning the units digit of 2 to the 57th power is 2. Hope that clears things up for you, sugar!
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What two digit number are interchanged to form a new two-digit number?
Any two digit number in which: (a) the units digit is not 0, and (b) the two digits are different will form a new 2-digit number when the digits are interchanged.
How many 2-digit integers greater than 50 have the property that the sum of the digits is more than double the units digit?
When the units digit equals the tens digit then the sum of the digits of a 2 digit number is double the units digit. In each tens range above 50, numbers below this critical point meet the requirement, numbers above this critical point have a sum LESS than double the units digit. The applicable numbers are 51-54 (4), 61-65 (5), 71-76 (6), 81-87 (7) and 91-98 (8). Then there are 4 + 5 + 6 + 7 + 8 = 30 qualifying numbers.
What is the units digit of the 5857th triangular number.?
The nth triangular number is given by ½ × n × (n+1)→ the 5857th triangular number is ½ × 5857 × 5858 = 17,155,153, so its units digit is a 3.------------------------------------------------------------Alternatively,If you look at the units digits of the first 20 triangular numbers they are {1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0}At this stage, as we are only concerned with the units digit, as we now have a 0 for the units digit, when 21 is added it is the same as adding 1 to 0 to give a 1, for the 22nd triangular number, we are adding 2 to the 1 to give 3, and so on - the sequence of 20 digits is repeating.To find the units digit of the nth triangular number, find the remainder of n divided by 20 and its units digit will be that digit in the sequence (if the remainder is 0, use the 20th number). To find the remainder when divided by 20 is very simple by looking at only the tens digit and the units digit:If the tens digit is even (ie one of {0, 2, 4, 6, 8}), the remainder is the units digitIf the tens digit is odd (ie one of {1, 3, 5, 7, 9}), the remainder is the units digit + 10.5857 ÷ 20 = ... remainder 17; the 17th digit of the above sequence is a 3, so the units digit of the 5857th triangular number is a 3.This trick can be used for much larger triangular numbers which calculators cannot calculate exactly using the above formula. eg the units digit of the 1234567890123456789th triangular number is... 1234567890123456789 ÷ 20 = .... remainder 9, so this triangular number's units digit is the 9th digit of the above sequence which is a 5.
Two prime numbers between 100 and 199 such that the tens digit is a prime number and the tens and ones digit taken together are a 2 digit prime number find the sum of these two prime numbers?
For the tens digit to be a prime number then it must equal 2, 3, 5 or 7. There are four 3-digit prime numbers that fit the above condition and also have the tens and units digits forming a 2-digit prime number. 131, 137, 173, 179. The person supplying the question may like to sum the various combinations.
What number is the square of its units digit?
Because the number is the square of its units digit, it must be the square of a number between 1 and 9. Look at the squares of the first nine numbers.The square of 1 is 1.The square of 2 is 4.The square of 3 is 9.The square of 4 is 16.The square of 5 is 25.The square of 6 is 36.The square of 7 is 49.The square of 8 is 64.The square of 9 is 81.Of these numbers, 1, 25, and 36 are squares of their units digit.
