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• Question - Magic Number 15

  Yossi writes the number `15` on the board. Then, Danny adds a digit to the right and a digit to the left of the number written on the board, such that the new number is still divisible by `15`.

  Find this number. Is there only one possibility?

  Note: The digit added to the left is not zero.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Proof and Example -> Constructing an Example / Counterexample Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25 Number Theory -> Division

• Question

  Answer without using pen and paper (and without a calculator, of course  ):

  Which number is larger: `12345678^2` or  `12345677*12345679` ?

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  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities

• Question

  Is it possible to make change for a `25` lira note using `10` coins worth `1`, `3`, and `5` lira?

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  Topics:

  Arithmetic Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd)

• Question

  All numbers from `45` to `65` are written on a board. How many times does the digit `5` appear?

  Topics:

  Number Theory Arithmetic

• Question

  Calculate without a calculator using the most efficient method:

  \( { (2001 \cdot 2021+100) \cdot (1991 \cdot 2031+400) \over 2011^4}\)

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  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities

• Question

  In two classes with an equal number of students, a quiz was administered. After grading the quiz, the teacher claimed that the number of grades of `0 ` was `13` greater than the number of all other grades combined. Is it possible that he was mistaken?

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  Topics:

  Combinatorics -> Double Counting Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction

• Question

  Can you find `35` integers whose average is equal to `6.35`?

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  Topics:

  Arithmetic Combinatorics -> Double Counting Algebra -> Inequalities -> Averages / Means Proof and Example -> Proof by Contradiction

• Question

  Given a three-digit prime number with all its digits distinct. It is known that its last digit is equal to the sum of the other two digits. Find all the possibilities for the last digit of this number.

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  Topics:

  Number Theory -> Prime Numbers Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 2, 4, and 8 Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  How can you divide `7` apples equally among `12` children, if you are not allowed to cut any apple into more than `5` pieces?

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  Topics:

  Algebra -> Word Problems Arithmetic -> Fractions Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division

• Question

  The journey from home to school takes Yossi `20` minutes. One day, when Yossi was already on his way to school, he remembered that he had forgotten his pen at home. If he continues to school now, he will arrive `3` minutes before the bell. And if he goes back home to get the pen, he will be `7 ` minutes late for class. What fraction of the way to school had Yossi traveled when he remembered that he didn't have a pen?

  Note: Yossi walks at a constant speed the entire time.

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  Topics:

  Arithmetic -> Fractions Algebra -> Word Problems -> Motion Problems