I am a DSP Engineer and a mathematician working in closely related areas of DSP, Digital Control, Digital Comm and Error Control Coding. I have a passion for both Pure and Applied Mathematics.

### Skill Check VII: IITJEE Foundation Math

A) Find the LCM of the following numbers by the prime factorization method: (a) 24, 36 and 72 (b) 84 and 112 (c) 144 and 192 (d) 624 and 520 (e) 225 and 270 (f) 1008 and 1512 (g) 2310, 1540, and 770 (h) 840, 504, and 672 (i) 528, 396, and 352 (j) 6552, 4368, and 9828.

B) Find the LCM of the following numbers by the common division method: (a) 336 and 224 (b) 840 and1260 (c) 630 and 840 and 504 (d) 864, 1296, and 576 (e) 144, 216, and 384 (f) 1764, 1176, and 2352 (g) 260, 390, 156, and 104 (h) 1170, 780, 1755, and 2340 (i) 2520, 1680, 3780, and 3024 (j) 2730, 1950, 3822, and 1820.

C) Find the smallest number that is exactly divisible by 2016 and 3024.

D) Find the greatest 5-digit number that is exactly divisible by 420, 490, and 280.

E) Find the smallest 6-digit number which, when divided by 96, 144, 72, and 192, leaves exactly 8 as a remainder.

F) The LCM of two coprime numbers is 70560. If one of the numbers is 245, find the other number.

G) The LCM of 42 and another number is 168. If the HCF of the two numbers is 14, find the other number.

H) Four bells begin to toll together. The bells tolls after 8, 10, 12, and 15 seconds, respectively. After how long will all four bells toll together?

I) A toy soldier salutes after taking 14 steps while another salutes after every 21 steps. If both toy soldiers take a step every second, how long will it take for the toy soldiers to salute together five times after starting off together?

Regards,

Nalin Pithwa

### Skill Check VI: IITJEE Foundation Math

A) Find all the factors of the following numbers: 42, 66, 88, 180, 810.

B) Use prime tree factorization to find the factors of : 1122, 2211, 2121, 8181, 8000.

C) Find the HCF of the following numbers by prime factorization: (a) 88 and 99 (b) 84 and 108 (c) 80 and 96 (d) 208 and 234

D) Find the HCF or GCD of the following numbers by Euclid’s Long Division Method: (a) 432, 540, and 648 (b) 408, 476, and 510 (c) 1350 and 1800 (d) 3600 and 5400 (e) 7560 and 8820 (f) 7920 and 8910 (g) 14112 and 12936 (h) 25740 and 24024 (i) 108, 288, and 360 (j) 1056, 1584, and 2178

E) Find the HCF or GCD of the following numbers by Euclid’s Long Division Method: (a) 1701, 1575, and 2016 (b) 4680, 4160, and 5200 (c) 3168. 3432, and 3696 (d) 4752, 5184, and 5616 (e) 8640, 10368, 12096 (f) 9072, 8400, 9744

F) Find the greatest number that divides 10368, 9504 and 11232 exactly leaving no remainders.

G) Find the greatest number that divides 7355, 8580, and 9805 leaving exactly 5 as a remainder in each case.

H) Find the greatest number that divides 9243 and 12325 leaving exactly 3 and 5 as remainders respectively.

I) What would be the length of the the longest tape that can be used to measure the length and breadth of an auditorium 204 feet wide and 486 feet long in an exact number of times.

J) A big cardboard picture 126 cm wide and 135 cm long is to be cut up into square pieces to create a jigsaw puzzle. How many small pieces would go on to make the jigsaw puzzle if each piece is to be equal and of the maximum possible size?

K) Square placards need to be cut out from a rectangular piece of cardboard 60 inches wide and 72 inches long. What is the maximum number of equal sized placards of the biggest possible size that can be cut out? What would be the length of each placard?

L) Three ribbons, 171 cm, 185 cm, and 199 cm long are to be cut into equal pieces of maximum possible length, leaving bits of ribbons 3 cm long from each. What would be the length of each piece of ribbon and how many such pieces can one get ?

M) The capacities of two emtpy water tanks are 504 litres and 490 litres. What would be the maximum capacity of a bucket that can be used an exact number of times to fill the tanks? How many bickets full of water will be needed?

Regards,

Nalin Pithwa

### Skill Check V: IITJEE Foundation Math

I. Consider the following relationship amongst various number systems: $\mathcal{N} \subset \mathcal{W} \subset \mathcal{Z} \subset \mathcal{Q} \subset \mathcal{R}$.

Write the following numbers in the smallest set or subset in the above relationship:

(a) 8 (b) -8 (c) +478 (d) -2191 (e) -21.91 (f) +3.6 (g) 0 (h) +4.6 (i) $-6.\dot{7}$ (j) 8.292992999…(k) $\frac{3}{8}$ (l) $\frac{8}{2}$ (m) $0 \frac{0}{7}$ (n) $-3\frac{1}{5}$ (o) $\frac{22}{33}$ (p) $\sqrt{64}$ (r) $\sqrt{6.4}$ (s) $2+\sqrt{3}$ (t) $6\sqrt{4}$ (u) $4\sqrt{6}$

II. If $\frac{22}{7} = 3.1428571...$ is $\frac{22}{7}$ an irrational number?

III. Fill in the boxes with the correct real numbers in the following statements: (a) $2\sqrt{7}+\sqrt{7} = \Box+ 2\sqrt{7}$ (b) $3.\dot{8} + 4.65 = 4.65 + \Box$ (c) $\Box + 29 = 29 + 5\sqrt{10}$ (d) $3.\dot{9} + (4.69 +2.12) = (\Box + 4.69) + 2.12$ (e) $(\frac{7}{8} + \frac{3}{7})+\frac{6}{5} = (\frac{6}{5} + \frac{3}{7}) + \Box$ (f) $3\sqrt{2}(\sqrt{3}+2\sqrt{5}) = (3\sqrt{2}+2\sqrt{5}) + (\Box \times \Box)$ (g) $1\frac{3}{7} (2\frac{7}{11} + \Box) = (1\frac{3}{7} \times 1 \frac{8}{9}) + (1\frac{3}{7} + 2\frac{7}{11})$ (h) $2\frac{1}{3} + \Box =0$ (xi) $\frac{7}{-8} \times \Box = 1$ (i) $-7.35 + \Box = 0$

IV. Find the answers to the following expressions by using the properties of addition and multiplication of real numbers:

Before that, we can recapitulate the relevant properties here :

Properties of Real Numbers:
Closure Property: The sum, difference, product,or quotient of two real numbers is a real number.

Commutative Property of Addition and Multiplication:

A change in the order of addition or multiplication of two real numbers does not change their respective sum or product. (a) $x+y = y+x$ (b) $x \times y = y \times x$

Associative Property of Addition and Multiplication:

A change in the grouping of three real numbers while adding or multiplying does not change their respective sum or product :

$(a+b)+c = (a+b)+c$ and $a \times (b \times c) = (a \times b) \times c$

Distributive Property of Multiplication over Addition:

When a real number is multiplied by the sum of two or more real numbers, the product is the same as the sum of the individual products of the real number and each addend.

$m(a+b) = ma+mb$. Clearly, multiplication has “distributed” over addition.

Identity Property of Real Numbers
The addition of zero or the multiplication with one does not change a real number. That is,

$a+0=0+a=a$ and $a \times 1 = a = 1 \times a$

Inverse Property of Real Numbers

• Corresponding to every real number, there exists another real number of opposite sign such that the sum of the two real numbers is zero: $a+ a^{'}=0$ such that $a^{'}=-a$
• Corresponding to every (non-zero) real number, there exists a real number, known as its reciprocal, such that the product of the two real numbers is 1. That is, $a \times \frac{1}{a} = 1$, where $r \neq 0$.

Now, in the questions below, identify the relevant properties:

(a) $283 +(717 + 386)$

(b) $(2154 - 1689) + 1689$

(c) $3.18 + (6.82+1.35)$

(d) $(6.784-3.297) + 3.297$

(e) $\frac{7}{13} + (\frac{6}{13}-1)$

(e) $0.25 \times (4.17 -0.17)$

(f) $(6.6 \times 6.6) + (6.6 \times 3.4)$

(g) $(\frac{2}{3} \times 5) - (\frac{2}{3} \times 2)$

(h) $(6.\dot{8} \times 5) - (6.\dot{8} \times 4)$

(i) $\frac{6}{7} \times \frac{7}{6} \times \frac{6}{7}$

V) Which of the following operations on irrational numbers are correct?

(a) $6\sqrt{5} - 4\sqrt{3}=2\sqrt{2}$

(b) $\sqrt{7} \times \sqrt{7} = 7$

(c) $3 \sqrt{3} + 3 \sqrt{3} = 6 \sqrt{3}$

(d) $\sqrt{7} \times \sqrt{7} = 49$

(e) $\sqrt{7} + \sqrt{2} = \sqrt{9}$

(f) $2 \sqrt{8} \times 3\sqrt{2} =24$

(g) $8\sqrt{2} + 8 \sqrt{2} =32$

(h) $2\sqrt{3}= 3\sqrt{6} = \frac{2}{3\sqrt{2}}$

(i) $5+\sqrt{3} = 5\sqrt{3}$

(j) $3\sqrt{20} \div 3\sqrt{5}=2$

VI) Find the rationalizing factors of the following irrational numbers:

(a) $\sqrt{10}$

(b) $\sqrt{7}$

(c) $2\sqrt{5}$

(d) $3\sqrt{7}$

(e) $-2\sqrt{8}$

(f) $-6\sqrt{7}$

(g) $\frac{1}{\sqrt{2}}$

(h) $\frac{2}{\sqrt{3}}$

(i) $2\sqrt{3}=4\sqrt{3}$

(j) $7\sqrt{5} - 2\sqrt{5}$

(k) $1+\sqrt{2}$

(l) $3-\sqrt{5}$

(m) $3\sqrt{2}+6$

(n) $4\sqrt{7} + 6\sqrt{2}$

(o) $3\sqrt{6}-2\sqrt{3}$

VII) Rationalize the denominators of the following numbers:

(a) $\frac{1}{\sqrt{3}}$

(b) $\frac{3}{\sqrt{3}}$

(c) $\frac{3}{\sqrt{5}}$

(d) $\frac{8}{\sqrt{6}}$

(e) $\frac{3}{2\sqrt{5}}$

(f) $\frac{\sqrt{5}}{\sqrt{7}}$

(g) $\frac{3\sqrt{3}}{3\sqrt{5}}$

(h) $\frac{3}{\sqrt{5}-sqrt{3}}$

(i) $\frac{5}{\sqrt{3}+\sqrt{2}}$

(j) $\frac{17}{4\sqrt{6}+3\sqrt{5}}$

(k) $\frac{3}{3+\sqrt{3}}$

(l) $\frac{11}{3\sqrt{5}-2\sqrt{3}}$

(m) $\frac{\sqrt{5}}{3\sqrt{5}-3\sqrt{2}}$

(n) $\frac{\sqrt{3}+1}{\sqrt{3}-1}$

(o) $\frac{\sqrt{5}-sqrt{2}}{\sqrt{5}+\sqrt{2}}$

VIII. Find the additive inverse of each of the following irrational numbers:

(i) $\sqrt{7}$ (ii) $3\sqrt{5}$ (iii) $-6\sqrt{7}$ (iv) $5+\sqrt{7}$ (v) $3\sqrt{7} - 2\sqrt{8}$

IX. Find the multiplicative inverse of each of the following irrational numbers:

(i) $\sqrt{6}$ (ii) $\frac{1}{2\sqrt{7}}$ (iii) $\frac{3\sqrt{8}}{2\sqrt{7}}$ (iv) $\frac{4}{3+\sqrt{2}}$ (v) $\frac{2\sqrt{5}+3\sqrt{6}}{5\sqrt{8}-4\sqrt{7}}$

X. Illustrate the closure property of addition of real numbers using the irrational numbers $\sqrt{5}$ and $2\sqrt{5}$.

XI. Illustrate that the closure property does not apply on subtraction of real numbers using two rational numbers: $2\frac{1}{7}$ and $-3\frac{2}{5}$.

XII. Illustrate the distributive property of multiplication over addition of real numbers using three irrational numbers: $3\sqrt{7}$, $-2\sqrt{7}$ and $\sqrt{7}$.

Regards,

Nalin Pithwa

### Set Theory Primer : Some basic thinking and problem solving

Reference: AMS, Student Mathematical Library: Basic Set Theory by A. Shen, et al. Chapter 1. Section 1.

Problem 1:

Consider the oldest mathematician amongst chess players and the oldest chess player amongst mathematicians. Could they be two different people?

Problem 2:

The same question for the best mathematician amongst chess players and the best chess player amongst mathematicians.

Problem 3:

One tenth of mathematicians are chess players, and one sixth of chess players are mathematicians. Which group (mathematicians or chess players) is bigger? What is the ratio of sizes of these two groups?

Problem 4:

Do there exist sets A, B and C such that $A \bigcap B \neq \phi$, $A \bigcap C = \phi$ and $(A\bigcap B)-C = \phi$ ?

Problem 5:

Which of the following formulas are true for arbitrary sets A, B and C:

i) $(A \bigcap B) \bigcup C = (A \bigcup C) \bigcap (B \bigcup C)$

ii) $(A \bigcup B) \bigcap C = (A \bigcap C) \bigcup (B \bigcap C)$

iii) $(A \bigcup B) - C = (A-C)\bigcup B$

iv) $(A \bigcap B) - C = (A - C) \bigcap B$

v) $A - (B \bigcup C) = (A-B) \bigcap (A-C)$

vi) $A - (B \bigcap C) = (A-B) \bigcup (A-C)$

Problem 6:

Give formal proofs of all valid formulas from the preceding problem. (Your proof should go like this : “We have to prove that the left hand side equals the right hand side. Let x be any element of the left hand side set. Then, ….Therefore, x belongs to the right hand side set. On the other hand, let…”)

Please give counterexamples to the formulas which are not true.

Problem 7:

Prove that the symmetric difference is associative:

$A \triangle (B \triangle C) = (A \triangle B) \triangle C$ for any sets A, B and C. Hint: Addition modulo two is associative.

Problem 8:

Prove that:

$(A_{1}\bigcap A_{2}\bigcap \ldots A_{n}) \triangle (B_{1} \bigcap B_{2} \bigcap \ldots B_{n}) = (A_{1} \triangle B_{1}) \bigcup (A_{2} \triangle B_{2}) \ldots (A_{n} \triangle B_{n})$ for arbitrary sets $A_{1}, A_{2}, \ldots, A_{n}, B_{1}, B_{2}, \ldots, B_{n}$.

Problem 9:

Consider an inequality whose left hand side and right hand side contain set variables and operations $\bigcap, \bigcup$ and -. Prove that if this equality is false for some sets, then it is false for some sets that contain at most one element.

Problem 10:

How many different expressions can be formed from set variables A and B by using union, intersection and set difference? (Variables and operations can be used more than once. Two expressions are considered identical if they assume the same value for each set of values of the variables involved.) Solve the same problem for three sets and for n sets. (Answer: In the general case, $2^{2^{n}-1}$)

Problem 11:

Solve the same problem if only $\bigcup$ and $\bigcap$ are allowed. For n=2 and n-3, this problem is easy to solve; however, no general formula for any n is known. This problem is also called “counting monotone Boolean functions in n variables”.)

Problem 12:

How many subsets does an n-element subset have?

Problem 13:

Assume that A consists of n elements and $B \subset A$ consists of k elements. Find the number of different sets C such that $B \subset C \subset A$.

Problem 14:

A set U contains 2n elements. We select k subsets of A in such a way that none of them is a subset of another one. What is the maximum possible value of k? (Hint: Maximal k is achieved when all subsets have n elements. Indeed, imagine the following process: We start with an empty set and add random elements one by one until we get U. At most one selected set can appear in this process. On the other hand, the expected number of selected sets that appear during this process can be computed using the linearity of expectation. Take into account that the probability to come across some set $Z \subset U$ is minimal when Z contains n elements, since all the sets of a given size are equiprobable.)

Regards,

Nalin Pithwa.

Purva building, 5A
Flat 06