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Explore the Back-StoryWrite down all the ten digit numbers whose digital sum is . (The digital sum of a number is the sum of the digits of the number. The digital sum of is is ). Find the sum of all the digit numbers with digital sum .

The sum of the -digit numbers and is divisible by . Find all possible pairs .

Three congruent circles with centres and , are tangent to the sides of rectangle as shown. The circle with centre at has diameter and passes through the points and . Find the area of the rectangle .

A lucky year is one in which at least one date, when written in the form day/month/year, has the following property. The product of the month times the day equals the last two digits of the year. For example, 1944 is a lucky year because it has the date where . From to how many years are not lucky ? Give proper explanation for your answer.

The area of each of the four congruent L-shaped regions of this by square is of the total area. How many centimeters long is the side of the centre square?

For any positive integer is the sum of the digits of . What is the minimum value of when (1) and (2) .

A digit number is obtained by writing the digit numbers to i.e., . You have to remove digits from this number in such a way that the remaining digits in that order form the largest number possible. (For example in if we remove the four 's we get the number , but if we remove and the after , we get . This will be the largest number possible in this case.) What will be the first 10 digits of the largest number obtained?

Given the numbers and is defined as the remainder when the ordinary product is divided by . Find the product of every pair of these numbers including the product of number with itself. Fill in the table given below.

(1) Find , where we find the product of fifteen .

(2) Find where we have ten

Write down all the ten digit numbers whose digital sum is . (The digital sum of a number is the sum of the digits of the number. The digital sum of is is ). Find the sum of all the digit numbers with digital sum .

The sum of the -digit numbers and is divisible by . Find all possible pairs .

Three congruent circles with centres and , are tangent to the sides of rectangle as shown. The circle with centre at has diameter and passes through the points and . Find the area of the rectangle .

A lucky year is one in which at least one date, when written in the form day/month/year, has the following property. The product of the month times the day equals the last two digits of the year. For example, 1944 is a lucky year because it has the date where . From to how many years are not lucky ? Give proper explanation for your answer.

The area of each of the four congruent L-shaped regions of this by square is of the total area. How many centimeters long is the side of the centre square?

For any positive integer is the sum of the digits of . What is the minimum value of when (1) and (2) .

A digit number is obtained by writing the digit numbers to i.e., . You have to remove digits from this number in such a way that the remaining digits in that order form the largest number possible. (For example in if we remove the four 's we get the number , but if we remove and the after , we get . This will be the largest number possible in this case.) What will be the first 10 digits of the largest number obtained?

Given the numbers and is defined as the remainder when the ordinary product is divided by . Find the product of every pair of these numbers including the product of number with itself. Fill in the table given below.

(1) Find , where we find the product of fifteen .

(2) Find where we have ten

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