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Qn 1. An integer N has digital representation a1a2a3. Moreover,
None of the digits a1, a2, or a3 and none of the numbers with digital
representation a1a2, a1a3, or a2a3 is divisible by 3.
N is odd.
N is divisible by 9.
a1 ≥ a2 ≥ a3.
Determine all possible numbers N.
given that n is a positive even integer, 5n + 4 will always be divisible
A. 5n
B.5
C. 2
D. 4
Three neon lights colored red, blue and green flash at different time intervals. The red light flashes after every 18 seconds and the green light after every 15 seconds. If all the three lights flash together at 8:00 am, how many times will all three lights flash together by 9:30 am.
Please show your full steps answer.
Find the last digit of the sum
0! + 2! + 4! +...+ 2010! + 2012!
Please show full steps answer.
Twelve cowboys sit in a circle around a bonfire. Each observes that his age (viewed as integer) is the average of the ages of his left and right neighbors. What is the sum of their ages?
Please show your answer with all complete steps.
a) i) Give an inductive formula for the sum of the first n odd numbers:
1 + 3 + 5 + ... + 2n -1
Show your induction process.
ii) Use the proof by mathematical induction to prove the correctness of your
inductive formula in i) above.
a) i) Give an inductive formula for the sum of the first n odd numbers:
1 + 3 + 5 + ... + 2n -1
Show your induction process.
ii) Use the proof by mathematical induction to prove the correctness of your
inductive formula in i) above.
Show that 224-1 and 216-1 are composite. Hints:use expansion of (a2-b2)
Show that any prime number is either in form of 4k+1 or 4k+3, k is any positive integer
Six professor begin courses on Monday, Tuesday, Wednesday, Thursday, friday, and Saturday, and announce their intention of lecturing at interval of 3,2,5,6,1and 4 days respectively the regulations of the university forbid Sunday a lecture(so that a Sunday a lecture must be omitted) when first will all six professors find themselves compelled to omit a lecture
