A,B,C is a right angled triangle with coordinates (-10,10); (0,10);(10,10). Please let me know how to find the number of integral cordinates that lie on/within the triangle.
Bull Mock Question
ABCD is an isosceles trapezium with parallel sides AB & CD having lengths of 6 cm and 12 cm respectively. Also AD = 6 cm. An in-circle is drawn inside ABCD with centre O. The area of the triangle OCD is (in sq. cm)
How many different numbers less than 100000 are there, such that the sum of their digits is 4?
An Useful Article on Number system i found on Bulls Eye Website.
Unit Digit of A Number
The concept that revolves around finding the unit digit of a number uses the basics of number system. Learning this concept means you have strengthened your basic concepts.
The concept of unit digit can be learned by figuring out the unit digits of all the single digit numbers from 0 - 9 when raised to certain powers. The first learning in that for you will be that these numbers can be broadly classified into three categories for this purpose:
Digits 0, 1, 5 & 6: When we observe the behavior of these digits, they all have the same unit's digit as the number itself when raised to any power, i.e. 0n = 0, 1n =1, 5n = 5, 6n = 6. So, it becomes simple to understand this logic.
Examples: Finding the Unit digit of following numbers:
185563 = 5; 2716987 = 1; 15625369 = 6; 190654789321 = 0.
Digits 4 & 9: Both these numbers are perfect squares and also have the same behavior with respect to their unit digits i.e. they have a cyclicity of only two different digits as their unit's digit.
Have a look at how the powers of 4 operate:
41 = 4, 42 = 16, 43 = 64 and so on.
Hence, the power cycle of 4 contains only 2 numbers 4 & 6, which appear in case of odd and even powers respectively.
Likewise 91 = 9, 92 = 81, 93 = 729 and so on.
Hence, the power cycle of 9 also contains only 2 numbers 9 & 1, which appear in case of odd and even powers respectively.
So broadly these can be remembered in even and odd only, i.e. 4odd = 4 and 4even = 6 and likewise 9odd = 9 and 9even = 1.
Examples: Finding the Unit digit of following numbers:
189562589743 = 9 (since power is odd); 279698745832 = 1(since power is even);
154258741369 = 4 (since power is odd); 19465478932 = 6 (since power is even).
Digits 2, 3, 7 & 8: These numbers have a power cycle of 4 different numbers
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n how many ways can 75 be written as a sum of n consecutive positive integers? (Hint : apply the concept of averages)
Mr. Balkrishnan can solve 12 Maths or 10 Physics problems in 18 min. and Mr. Jaiswal can solve 6 Maths or 5 Physics problems in 24 min. They had to solve 20 Maths and 50 Physics problems. The time required when they work together is
[if gte mso 9]>123456789123456789............ upto 180 digits when divided by 11 what would be the remainder.
Please answer with explanation.
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Domain of Mod of root x ?
a. x>0
b. x>/ 0
12345678910111213141516171819....... upto 400 is when divided by 9 what would be the remainder. Answer with approach.
How many ordered natural numbers (J,K) are there such that (J - K) = 3 and difference in the number of zeroes of J! and K! = 2, given that both J, K are less than 300.
How many two-digit prime numbers have their digital root equal to a prime number?
x+y
x and y are positive integers.... x is not equal to y.
how many different solutions of x+y are possible..
find the number of integral solutions for x*y*z=30
What percent of all 5 digit natural numbers, whose sum of digits is 43, are divisible by 11 ?
Four distinct non-zero single digit numbers a, b, c and d satisfy the following four conditions:
I. The sum of (b, c, d) is a multiple of 9.
II. The sum of (a, c, d) is a multiple of 8.
III. The sum of (a, b, d) is a multiple of 5.
IV. The sum of (a, b, c) is a multiple of 23.
which is the largest number?
in how many ways can 7^11 be expressed as a product of three factors?
what is the remainder when 12345678987654321 divided by 1001 and 999?Please mention stepwise solution.