For an integer n , let S n = n + 1 , n + 2 , … . , n + 18 . Which of the following is true for all n ≥ 10 ?

S n has at least four multiples of 5

MEDIUM

Earn 100

$1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$

$11$ $12$ $13$ $14$ $15$ $16$ $17$ $18$ $19$ $20$

$21$ $22$ $23$ $24$ $25$ $26$ $27$ $28$ $29$ $30$

$31$ $32$ $33$ $34$ $35$ $36$ $37$ $38$ $39$ $40$

$41$ $42$ $43$ $44$ $45$ $46$ $47$ $48$ $49$ $50$

$51$ $52$ $53$ $54$ $55$ $56$ $57$ $58$ $59$ $60$

$61$ $62$ $63$ $64$ $65$ $66$ $67$ $68$ $69$ $70$

$71$ $72$ $73$ $74$ $75$ $76$ $77$ $78$ $79$ $80$

$81$ $82$ $83$ $84$ $85$ $86$ $87$ $88$ $89$ $90$

$91$ $92$ $93$ $94$ $95$ $96$ $97$ $98$ $99$ $100$

Note down the prime numbers between $1$ to $100$ derived from Sieve of Eratosthenes method.

Important Questions on Playing with Numbers

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

Find the smallest positive integer

n \geq 10

such that

n + 6

is a prime and

9 n + 7

is a perfect square.

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

A two-digit number

\bar{a b}

is called almost prime if one obtains a two-digit prime number by changing at most one of its digits

a & b

. (For example, 18 is an almost prime number because 13 is a prime number). Then the number of almost prime two-digit numbers is

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

There are four prime numbers written in ascending order. The product of the first three is

385

and that of the last three is

1001 .

The last number is :

HARD

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

The prime numbers

a, b

and

c

are such that

a + b^{2} = 4 c^{2} .

Determine the sum of all possible values of

a + b + c .

MEDIUM

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

For an integer

n

, let

S_{n} = \{n + 1, n + 2, \dots ., n + 18\}

. Which of the following is true for all

n \geq 10

HARD

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

Let

N

denote the number of all natural numbers

n

such that

n

is divisible by a prime

p > \sqrt{n}

and

p < 20 .

What is the value of

N ?

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

A prime number

p

is called special if there exist primes

p_{1}, p_{2}, p_{3}, p_{4}

such that

p = p_{1} + p_{2} = p_{3} - p_{4} .

The number of special primes is

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

Express

120

as a product of its prime factors.

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

Which of the following number is an odd and prime number?

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

The sum of the squares of the digits of the largest two-digit prime number is:

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

a

and

b

are positive integers such that

a^{2} - b^{2}

is a prime number, then

a^{2} - b^{2}

is equal to:

HARD

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

The number of

6

-digit numbers of the form

a b a b a b

(in base

10

) each of which is a product of exactly

6

distinct primes is

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

It is given that the number

43361

can be written as a product of two distinct prime numbers

p_{1}, p_{2} .

Further, assume that there are

42900

numbers which are less than

43361

and are co-prime to it. Then,

p_{1} + p_{2}

MEDIUM

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

Consider the set

E = {5, 6, 7, 8, 9} .

For any partition

{A, B}

E,

with both

A

and

B

non-empty, consider the number obtained by adding the product of elements of

A

to the product of elements of

B .

Let

N

be the largest prime number among these numbers. Find the sum of the digits of

N .

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

Express

120

as a product of its prime factors.

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

The number of prime numbers which are less than

100

is:

HARD

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

For

a, b \in \{0, 1, \dots, 9\}

, a two digit number

a b

is defined for

10 a + b

. A two digit number

a b

is called a special prime if both

a b

and

b a

are primes. Then the total number of special primes is

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

Which of the following is not a pair of prime numbers ?

HARD

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

Suppose

a, b, c

are positive integers such that

2^{a} + 4^{b} + 8^{c} = 328 .

Then

\frac{a + 2 b + 3 c}{a b c}

is equal to-

EASY

Mathematics>Arithmetic>Playing with Numbers>Prime and Composite Numbers

a

and

b

are positive integers such that

(a^{2} - b^{2})

is a prime number, then

$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$11$	$12$	$13$	$14$	$15$	$16$	$17$	$18$	$19$	$20$
$21$	$22$	$23$	$24$	$25$	$26$	$27$	$28$	$29$	$30$
$31$	$32$	$33$	$34$	$35$	$36$	$37$	$38$	$39$	$40$
$41$	$42$	$43$	$44$	$45$	$46$	$47$	$48$	$49$	$50$
$51$	$52$	$53$	$54$	$55$	$56$	$57$	$58$	$59$	$60$
$61$	$62$	$63$	$64$	$65$	$66$	$67$	$68$	$69$	$70$
$71$	$72$	$73$	$74$	$75$	$76$	$77$	$78$	$79$	$80$
$81$	$82$	$83$	$84$	$85$	$86$	$87$	$88$	$89$	$90$
$91$	$92$	$93$	$94$	$95$	$96$	$97$	$98$	$99$	$100$

Important Questions on Playing with Numbers

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