Math moments: uses of mathematics in today’s world

# motivation for Math

# Questions based on Wilson’s theorem for training for RMO

1(a) Find the remainder when is divided by 17.

1(b) Find the remainder when is divided by 29.

2: Determine whether 17 is a prime by deciding if

3: Arrange the integers 2,3,4, …, 21 in pairs a and b that satisfy .

4: Show that .

5a: Prove that an integer is prime if and only if .

5b: If n is a composite integer, show that , except when .

6: Given a prime number p, establish the congruence

7: If p is prime, prove that for any integer a, and

8: Find two odd primes for which the congruence holds.

9: Using Wilson’s theorem, prove that for any odd prime p:

10a: For a prime p of the form , prove that either

or

10b: Use the part (a) to show that if is prime, then the product of all the even integers less than p is congruent modulo p to either 1 or -1.

More later,

Nalin Pithwa.

# Eight digit bank identification number and other problems of elementary number theory

Question 1:

Consider the eight-digit bank identification number , which is followed by a ninth check digit chosen to satisfy the congruence

(a) Obtain the check digits that should be appended to the two numbers 55382006 and 81372439.

(b) The bank identification number has an illegitimate fourth digit. Determine the value of the obscured digit.

Question 2:

(a) Find an integer having the remainders 1,2,5,5 when divided by 2, 3, 6, 12 respectively (Yih-hing, died 717)

(b) Find an integer having the remainders 2,3,4,5 when divided by 3,4,5,6 respectively (Bhaskara, born 1114)

(c) Find an integer having remainders 3,11,15 when divided by 10, 13, 17, respectively (Regiomontanus, 1436-1476)

Question 3:

Question 3:

Let denote the nth triangular number. For which values of n does divide

Hint: Because , it suffices to determine those n satisfying

Question 4:

Find the solutions of the system of congruences:

Question 5:

Obtain the two incongruent solutions modulo 210 of the system

Question 6:

Use Fermat’s Little Theorem to verify that 17 divides

Question 7:

(a) If , show that . Hint: From Fermat’s Little Theorem, and

(b) If , show that divides

(c) If , show that

Question 8:

Show that and . Do there exist infinitely many composite numbers n with the property that and ?

Question 9:

Prove that any integer of the form is an absolute pseudoprime if all three factors are prime; hence, is an absolute pseudoprime.

Question 10:

Prove that the quadratic congruence , where p is an odd prime, has a solution if and only if .

Note: By quadratic congruence is meant a congruence of the form with . This is the content of the above proof.

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Nalin Pithwa.

# Pre RMO algebra : some tough problems

Question 1:

Find the cube root of

Question 2:

Find the square root of

Question 3:

Simplify (a):

Simplify (b):

Question 4:

Solve :

Question 5:

Solve the following simultaneous equations:

and

Question 6:

Simplify (a):

Simplify (b):

Question 7:

Find the HCF and LCM of the following algebraic expressions:

and and

Question 8:

Simplify the following using two different approaches:

Question 9:

Solve the following simultaneous equations:

Slatex x^{2}y^{2} + 192 = 28xy$ and

Question 10:

If a, b, c are in HP, then show that

Question 11:

if , prove that

Question 12:

Determine the ratio if we know that

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Nalin Pithwa

Those interested in such mathematical olympiads should refer to:

https://olympiads.hbcse.tifr.res.in

(I am a tutor for such mathematical olympiads).

# Elementary Number Theory, ISBN numbers and mathematics olympiads

Question 1:

The International Standard Book Number (ISBN) used in many libraries consists of nine digits followed by a tenth check digit (somewhat like Hamming codes), which satisfies

Determine whether each of the ISBN’s below is correct.

(a) 0-07-232569-0 (USA)

(b) 91-7643-497-5 (Sweden)

(c) 1-56947-303-10 (UK)

Question 2:

When printing the ISBN , two unequal digits were transposed. Show that the check digits detected this error.

Remark: Such codes are called error correcting codes and are fundamental to wireless communications including cell phone technologies.

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Nalin Pithwa.

# Mathematics Olympiads: A curious calculation and its cute proof !!

Explain why the following calculations hold:

Hint:

Show that

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Nalin Pithwa