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Let a be an integer such that a+1 is divisible by 3. Prove that 4+7a is divisible by 3.
Fred has five cards labeled 1, 2, 3, 4, and 5. Create a three-digit number and a two-digit number using these cards so that the larger number is divisible by the smaller one.
The diagram shows a point E inside a rectangle ABCD such that AE = 16 cm, DE = 20 cm and CE = 13 cm. Find the length of BE.
Find the missing number in the grid below.
Hint: Four-digit perfect squares.
What is the largest product that can be formed from using the digits 1, 2, 3, and 4, and one multiplication sign? You are only allowed to combine the digits to form two numbers.
Except for the first term, each term in the following sequence is found by multiplying the same number to the previous term:3, a, b, c, 48, …Find the value of c.
On December 31st, Michael made a statement: "After the New Year, everything I said before the New Year will become a lie." Was he telling the truth?
Santa Claus gave out 47 chocolates to a group of kids. Every girl got one more chocolate than every boy. After that, Santa shared 74 jelly beans, ensuring each boy got one more jelly bean than each girl. How many kids were there in total?
How many different ways are there to read the word BANANA in the following table if we can only cross to a field that shares an edge with the current field, and we can use fields several times?
A number is chosen at random from among the first 100 positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by 11?
A quadrilateral has all integer side lengths, a perimeter of 26, and one side of length � n. What is the greatest possible length of one side of this quadrilateral?
There are two hourglasses, one measuring 7 minutes and the other 11 minutes. You need to boil an egg for exactly 15 minutes. How can you measure this time using only the given hourglasses?
Kate has a calculator with three operations: multiply a number by 3, add 3 to a number, or (if the number is divisible by 3) divide it by 3. Using these operations, how can you transform the number 1 into the number 11?
A mother has two apples, three pears, and four oranges. For nine consecutive days, she gives her son one piece of the remaining fruit each day. In how many ways can she give the fruits to her son over the nine days?
On a globe, there are 17 parallels of latitude and 24 meridians of longitude. How many regions does the surface of the globe get divided into by the intersection of these parallels and meridians?
Prove that the number of USA states with an odd number of neighbours is even.
Define an unfortunate number as a number which is 13 times the sum of its digits. How many unfortunate numbers are there?
Replace the '*' in the expression ∗1∗2∗4∗8∗16∗32∗64=27 with either '+' or '−' to make the equation correct.
In how many ways can we cut a necklace consisting of 30 different beads into 8 parts? (It is allowed to cut only between beads.)
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