1. Find the smallest number which should be added to the largest 7-digit number to make it exactly divisible by 23.
The largest 7-digit number is 9,999,999. When divided by 23, the remainder is 18. To make it divisible by 23, add 23 – 18 = 5. So, 5 should be added.
2. Mark three non-collinear points X, Y, Z. Draw lines through these points taking two at a time. Name the lines. How many such different lines can be drawn?
Three lines can be drawn: XY, YZ, and ZX.
3. How many degrees are there in 4/5 of a right angle?
A right angle is 90°. Thus, 4/5 of a right angle is (4/5)*90° = 72°.
4. Find the greatest number which exactly divides 134, 178, and 262.
The greatest common divisor (GCD) of 134, 178, and 262 is 2.
5. Find the greatest 4-digit number which is exactly divisible by 19, 29, and 39.
To find the greatest 4-digit number divisible by 19, 29, and 39, we first need to determine the least common multiple (LCM) of these numbers.
However, notice that 39 is divisible by 3 and 13. So, if a number is divisible by both 13 and 3, it will automatically be divisible by 39.
Now, let’s find the LCM of 19, 29, and 13:
The greatest 4-digit number is 9,999. To find the greatest 4-digit number divisible by 7,223, divide 9,999 by 7,223 to get the quotient.
9,999 ÷ 7,223 = 1 (remainder = 2,776)
So, the largest 4-digit number divisible by 7,223 is 9,999 – 2,776 = 7,223 (which is the LCM itself).
Therefore, the greatest 4-digit number that is exactly divisible by 19, 29, and 39 is 7,223.
However, notice that 39 is divisible by 3 and 13. So, if a number is divisible by both 13 and 3, it will automatically be divisible by 39.
Now, let’s find the LCM of 19, 29, and 13:
- The prime factorization of 19 is 19 (itself, since it’s prime).
- The prime factorization of 29 is 29 (itself, since it’s prime).
- The prime factorization of 13 is 13 (itself, since it’s prime).
The greatest 4-digit number is 9,999. To find the greatest 4-digit number divisible by 7,223, divide 9,999 by 7,223 to get the quotient.
9,999 ÷ 7,223 = 1 (remainder = 2,776)
So, the largest 4-digit number divisible by 7,223 is 9,999 – 2,776 = 7,223 (which is the LCM itself).
Therefore, the greatest 4-digit number that is exactly divisible by 19, 29, and 39 is 7,223.
6. Find the least 6-digit number which is exactly divisible by 12, 36, and 48.
The least common multiple (LCM) of 12, 36, and 48 is 144.
The smallest 6-digit number is 100,000.
When 100,000 is divided by 144, the remainder is 64.
To make it divisible by 144, we add 144 – 64 = 80.
So, the least 6-digit number divisible by 12, 36, and 48 is 100,080.
7. Find the greatest 5-digit number which is exactly divisible by 7, 8, 9, and 11.
The least common multiple (LCM) of 7, 8, 9, and 11 is 5544.
The greatest 5-digit number is 99,999.
When 99,999 is divided by 5544, the remainder is 5039.
Subtracting this remainder from 99,999, we get 94,960.
So, the greatest 5-digit number divisible by 7, 8, 9, and 11 is 94,960.
8. Find the least number, which when divided by 10, 20, and 40 leaves a remainder 5.
The least common multiple (LCM) of 10, 20, and 40 is 40.
A number that leaves a remainder of 5 when divided by these numbers will be LCM + 5.
So, the least number that satisfies the given condition is 40 + 5 = 45.
9. Find the greatest number which when divides 1179 and 1498 leaves a remainder of 3 and 2 respectively.
The numbers obtained by subtracting the remainders from the given numbers are 1176 and 1496.
The greatest number which divides these two adjusted numbers is their greatest common divisor (GCD).
The GCD of 1176 and 1496 is 8.
So, the greatest number that satisfies the given condition is 8.
10. A rectangular room is 2400 cm long and 1800 cm broad. It is to be paved with square tiles of the same size. Find the least possible number of such tiles.
To find the side of the largest square tile that can be used, we need to find the GCD of 2400 and 1800.
GCD of 2400 and 1800 is 600.
Thus, the side of the largest square tile that can be used is 600 cm.
Area of each tile = 600 x 600 = 360,000 sq.cm.
Area of the room = 2400 x 1800 = 4,320,000 sq.cm.
Number of tiles required = 4,320,000/360,000 = 12.
So, 12 tiles are required.