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.
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