Page 1 1. The sum of a natural number and its square is 156. Find the number. Sol: Let the required natural number be x. According to the given condition, 2 2 2 156 156 0 13 12 156 0 13 12 13 0 13 12 0 13 0 12 0 13 12 x x x x x x x x x x x x x or x x or x 12 x ( x cannot be negative) Hence, the required natural number is 12. 2. The sum of natural number and its positive square root is 132. Find the number. Sol: Let the required natural number be x. According to the given condition, 132 x x Putting 2 , x y o r x y we get 2 2 2 132 132 0 12 11 132 0 12 11 12 0 12 11 0 12 0 11 0 12 11 y y y y y y y y y y y y y or y y or y 11 y ( y cannot be negative) Now, 2 11 11 121 x x Hence, the required natural number is 121. Page 2 1. The sum of a natural number and its square is 156. Find the number. Sol: Let the required natural number be x. According to the given condition, 2 2 2 156 156 0 13 12 156 0 13 12 13 0 13 12 0 13 0 12 0 13 12 x x x x x x x x x x x x x or x x or x 12 x ( x cannot be negative) Hence, the required natural number is 12. 2. The sum of natural number and its positive square root is 132. Find the number. Sol: Let the required natural number be x. According to the given condition, 132 x x Putting 2 , x y o r x y we get 2 2 2 132 132 0 12 11 132 0 12 11 12 0 12 11 0 12 0 11 0 12 11 y y y y y y y y y y y y y or y y or y 11 y ( y cannot be negative) Now, 2 11 11 121 x x Hence, the required natural number is 121. 3. The sum of two natural number is 28 and their product is 192. Find the numbers. Sol: Let the required number be x and 28 . x According to the given condition, 2 2 2 28 192 28 192 28 192 0 16 12 192 0 16 12 16 0 12 16 0 12 0 16 0 x x x x x x x x x x x x x x x or x 12 16 x o r x When 12, x 28 28 12 16 x When 16, x 28 28 16 12 x Hence, the required numbers are 12 and 16. 4. The sum of the squares of two consecutive positive integers is 365. Find the integers. Sol: Let the required two consecutive positive integers be x and 1 . x According to the given condition, 2 2 2 2 2 2 2 1 365 2 1 365 2 2 364 0 182 0 14 13 182 0 x x x x x x x x x x x x 14 13 14 0 14 13 0 14 0 13 0 14 13 x x x x x x or x x or x 13 x ( x is a positive integers) When 13, x 1 13 1 14 x Page 3 1. The sum of a natural number and its square is 156. Find the number. Sol: Let the required natural number be x. According to the given condition, 2 2 2 156 156 0 13 12 156 0 13 12 13 0 13 12 0 13 0 12 0 13 12 x x x x x x x x x x x x x or x x or x 12 x ( x cannot be negative) Hence, the required natural number is 12. 2. The sum of natural number and its positive square root is 132. Find the number. Sol: Let the required natural number be x. According to the given condition, 132 x x Putting 2 , x y o r x y we get 2 2 2 132 132 0 12 11 132 0 12 11 12 0 12 11 0 12 0 11 0 12 11 y y y y y y y y y y y y y or y y or y 11 y ( y cannot be negative) Now, 2 11 11 121 x x Hence, the required natural number is 121. 3. The sum of two natural number is 28 and their product is 192. Find the numbers. Sol: Let the required number be x and 28 . x According to the given condition, 2 2 2 28 192 28 192 28 192 0 16 12 192 0 16 12 16 0 12 16 0 12 0 16 0 x x x x x x x x x x x x x x x or x 12 16 x o r x When 12, x 28 28 12 16 x When 16, x 28 28 16 12 x Hence, the required numbers are 12 and 16. 4. The sum of the squares of two consecutive positive integers is 365. Find the integers. Sol: Let the required two consecutive positive integers be x and 1 . x According to the given condition, 2 2 2 2 2 2 2 1 365 2 1 365 2 2 364 0 182 0 14 13 182 0 x x x x x x x x x x x x 14 13 14 0 14 13 0 14 0 13 0 14 13 x x x x x x or x x or x 13 x ( x is a positive integers) When 13, x 1 13 1 14 x Hence, the required positive integers are 13 and 14. 5. The sum of the squares to two consecutive positive odd numbers is 514. Find the numbers. Sol: Let the two consecutive positive odd numbers be x and 2 . x According to the given condition, 2 2 2 2 2 2 2 2 514 4 4 514 2 4 510 0 2 255 0 17 15 255 0 17 15 17 0 17 15 0 17 0 15 0 17 15 x x x x x x x x x x x x x x x x x x or x x or x 15 x ( x is a positive odd number) When 15, x 2 15 2 17 x Hence, the required positive integers are 15 and 17. 6. The sum of the squares of two consecutive positive even numbers is 452. Find the numbers. Sol: Let the two consecutive positive even numbers be x and 2 . x According to the given condition, 2 2 2 452 x x 2 2 4 4 452 x x x 2 2 4 448 0 x x 2 2 224 0 x x 2 16 14 224 0 x x x 16 14 16 0 16 14 0 16 0 14 0 16 14 x x x x x x o r x x o r x 14 x ( x is a positive even number) When 14, x 2 14 2 16 x Page 4 1. The sum of a natural number and its square is 156. Find the number. Sol: Let the required natural number be x. According to the given condition, 2 2 2 156 156 0 13 12 156 0 13 12 13 0 13 12 0 13 0 12 0 13 12 x x x x x x x x x x x x x or x x or x 12 x ( x cannot be negative) Hence, the required natural number is 12. 2. The sum of natural number and its positive square root is 132. Find the number. Sol: Let the required natural number be x. According to the given condition, 132 x x Putting 2 , x y o r x y we get 2 2 2 132 132 0 12 11 132 0 12 11 12 0 12 11 0 12 0 11 0 12 11 y y y y y y y y y y y y y or y y or y 11 y ( y cannot be negative) Now, 2 11 11 121 x x Hence, the required natural number is 121. 3. The sum of two natural number is 28 and their product is 192. Find the numbers. Sol: Let the required number be x and 28 . x According to the given condition, 2 2 2 28 192 28 192 28 192 0 16 12 192 0 16 12 16 0 12 16 0 12 0 16 0 x x x x x x x x x x x x x x x or x 12 16 x o r x When 12, x 28 28 12 16 x When 16, x 28 28 16 12 x Hence, the required numbers are 12 and 16. 4. The sum of the squares of two consecutive positive integers is 365. Find the integers. Sol: Let the required two consecutive positive integers be x and 1 . x According to the given condition, 2 2 2 2 2 2 2 1 365 2 1 365 2 2 364 0 182 0 14 13 182 0 x x x x x x x x x x x x 14 13 14 0 14 13 0 14 0 13 0 14 13 x x x x x x or x x or x 13 x ( x is a positive integers) When 13, x 1 13 1 14 x Hence, the required positive integers are 13 and 14. 5. The sum of the squares to two consecutive positive odd numbers is 514. Find the numbers. Sol: Let the two consecutive positive odd numbers be x and 2 . x According to the given condition, 2 2 2 2 2 2 2 2 514 4 4 514 2 4 510 0 2 255 0 17 15 255 0 17 15 17 0 17 15 0 17 0 15 0 17 15 x x x x x x x x x x x x x x x x x x or x x or x 15 x ( x is a positive odd number) When 15, x 2 15 2 17 x Hence, the required positive integers are 15 and 17. 6. The sum of the squares of two consecutive positive even numbers is 452. Find the numbers. Sol: Let the two consecutive positive even numbers be x and 2 . x According to the given condition, 2 2 2 452 x x 2 2 4 4 452 x x x 2 2 4 448 0 x x 2 2 224 0 x x 2 16 14 224 0 x x x 16 14 16 0 16 14 0 16 0 14 0 16 14 x x x x x x o r x x o r x 14 x ( x is a positive even number) When 14, x 2 14 2 16 x Hence, the required numbers are 14 and 16. 7. The product of two consecutive positive integers is 306. Find the integers. Sol: Let the two consecutive positive integers be x and 1 . x According to the given condition, 2 2 1 306 306 0 18 17 306 0 18 17 18 0 18 17 0 18 0 17 0 18 17 x x x x x x x x x x x x x or x x or x 17 x ( x is a positive integers) When 17, x 1 17 1 18 x Hence, the required integers are 17 and 18. 8. Two natural number differ by 3 and their product is 504. Find the numbers. Sol: Let the required numbers be x and 3 . x According to the question: 3 504 x x 2 3 504 x x 2 3 504 0 x x 2 24 21 504 0 x x 2 24 21 504 0 x x x 24 21 24 0 x x x 24 21 0 24 0 21 0 24 21 x x x o r x x o r x If 24, x the numbers are 24 and 24 3 21 . If 21, x the numbers are 21 and 21 3 24 . Hence, the numbers are 24, 21 and 21,24 . Page 5 1. The sum of a natural number and its square is 156. Find the number. Sol: Let the required natural number be x. According to the given condition, 2 2 2 156 156 0 13 12 156 0 13 12 13 0 13 12 0 13 0 12 0 13 12 x x x x x x x x x x x x x or x x or x 12 x ( x cannot be negative) Hence, the required natural number is 12. 2. The sum of natural number and its positive square root is 132. Find the number. Sol: Let the required natural number be x. According to the given condition, 132 x x Putting 2 , x y o r x y we get 2 2 2 132 132 0 12 11 132 0 12 11 12 0 12 11 0 12 0 11 0 12 11 y y y y y y y y y y y y y or y y or y 11 y ( y cannot be negative) Now, 2 11 11 121 x x Hence, the required natural number is 121. 3. The sum of two natural number is 28 and their product is 192. Find the numbers. Sol: Let the required number be x and 28 . x According to the given condition, 2 2 2 28 192 28 192 28 192 0 16 12 192 0 16 12 16 0 12 16 0 12 0 16 0 x x x x x x x x x x x x x x x or x 12 16 x o r x When 12, x 28 28 12 16 x When 16, x 28 28 16 12 x Hence, the required numbers are 12 and 16. 4. The sum of the squares of two consecutive positive integers is 365. Find the integers. Sol: Let the required two consecutive positive integers be x and 1 . x According to the given condition, 2 2 2 2 2 2 2 1 365 2 1 365 2 2 364 0 182 0 14 13 182 0 x x x x x x x x x x x x 14 13 14 0 14 13 0 14 0 13 0 14 13 x x x x x x or x x or x 13 x ( x is a positive integers) When 13, x 1 13 1 14 x Hence, the required positive integers are 13 and 14. 5. The sum of the squares to two consecutive positive odd numbers is 514. Find the numbers. Sol: Let the two consecutive positive odd numbers be x and 2 . x According to the given condition, 2 2 2 2 2 2 2 2 514 4 4 514 2 4 510 0 2 255 0 17 15 255 0 17 15 17 0 17 15 0 17 0 15 0 17 15 x x x x x x x x x x x x x x x x x x or x x or x 15 x ( x is a positive odd number) When 15, x 2 15 2 17 x Hence, the required positive integers are 15 and 17. 6. The sum of the squares of two consecutive positive even numbers is 452. Find the numbers. Sol: Let the two consecutive positive even numbers be x and 2 . x According to the given condition, 2 2 2 452 x x 2 2 4 4 452 x x x 2 2 4 448 0 x x 2 2 224 0 x x 2 16 14 224 0 x x x 16 14 16 0 16 14 0 16 0 14 0 16 14 x x x x x x o r x x o r x 14 x ( x is a positive even number) When 14, x 2 14 2 16 x Hence, the required numbers are 14 and 16. 7. The product of two consecutive positive integers is 306. Find the integers. Sol: Let the two consecutive positive integers be x and 1 . x According to the given condition, 2 2 1 306 306 0 18 17 306 0 18 17 18 0 18 17 0 18 0 17 0 18 17 x x x x x x x x x x x x x or x x or x 17 x ( x is a positive integers) When 17, x 1 17 1 18 x Hence, the required integers are 17 and 18. 8. Two natural number differ by 3 and their product is 504. Find the numbers. Sol: Let the required numbers be x and 3 . x According to the question: 3 504 x x 2 3 504 x x 2 3 504 0 x x 2 24 21 504 0 x x 2 24 21 504 0 x x x 24 21 24 0 x x x 24 21 0 24 0 21 0 24 21 x x x o r x x o r x If 24, x the numbers are 24 and 24 3 21 . If 21, x the numbers are 21 and 21 3 24 . Hence, the numbers are 24, 21 and 21,24 . 9. Find two consecutive multiples of 3 whose product is 648. Sol: Le the required consecutive multiples of 3 be 3 x and 3 1 . x According to the given condition, 2 2 2 2 3 3 1 648 9 648 72 72 0 9 8 72 0 9 8 9 0 9 8 0 9 0 8 0 9 8 x x x x x x x x x x x x x x x x x or x x or x 8 x (Neglecting the negative value) When 8, x 3 3 8 24 3 1 3 8 1 3 9 27 x x Hence, the required multiples are 24 and 27. 10. Find the tow consecutive positive odd integer whose product s 483. Sol: Let the two consecutive positive odd integers be x and 2 . x According to the given condition, 2 483 x x 2 2 483 0 x x 2 23 21 483 0 x x x 23 21 23 0 x x x 23 21 0 23 0 21 0 23 21 x x x or x x o r x 21 x ( x is a positive odd integer) When 21, x 2 21 2 23 x Hence, the required integers are 21 and 23.Read More

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