1
PREVIOUS YEARS SCHOOL SUMMATIVE-EXAM QUESTIONS
( REAL NUMBERS)
Q1-H.C.F. of two consecutive even numbers is :
(A) 0 (B) 1 (C) 4 (D) 2
Q2-If the HCF of 85 and 153 is expressible in the form 85n - 153, then value of n is :
(A) 3 (B) 2 (C) 4 (D) 1
Q3-If the HCF of 65 and 117 is expressible in the form 65 m-117, then the value of m
is :
(A) 4 (B) 2 (C) 3 (D) 1
Q4-Rational number , q 0 will be terminating decimal if the prime factorisation of
q is of the form. (m and n are non negative integers :
(A) 2m x3n (B) 2mx 5n (C) 3m x 5n (D) 3m x7n
Q5-For any two positive integers a and b, there exist unique integers q and r such that
a=bq+r, 0
If b =4 then which is not the value of r ?
(A) 1 (B) 2 (C) 3 (D) 4
Q6-The decimal expansion of
(A) 1 (B) 2 (C) 3 (D) 4
will terminate after how many places of decimal ?
Q7-Which of the following rational numbers has non terminating and repeating
decimal expansion
(A)
(B)
(C)
(D)
Q8-The decimal expansion of
(A) 1 (B) 2 (C) 3 (D) 4
will terminate after how many places of decimal ?
Q9-The decimal expansion of will terminate after how many places of decimals ?
(A) 2 (B) 1 (C) 3 (D) will not terminate
Q10-For some integer ‘m’ every odd integer is of the form :
(A) m (B) m+1 (C) 2 m (D) 2 m+1
Q11-If two positive integers A and B can be expressed as A= ab2 and B = a3b, where a,
b, are prime numbers, then LCM (A, B) is :
(A) ab (B) a2b2 (C) a3b2 (D) a4b3
2
Q12-Decimal expansion of
is :
(A) 0.115 (B) 1.15 (C) 11.5 (D) 0.0115
Q13-Which of the following numbers has terminating decimal expansion ?
(A) (B)
(C)
(D)
Q14-Which of the following rational numbers have a terminating decimal expansion?
(A) )
(B) )
(C) )
(D)
Q15-The decimal expansion of
decimals ?
(A) 2 (B) 3 (C) 4 (D) 5
will terminate after how many places of
Q16-Given that HCF (2520, 6600) = 40, LCM (2520, 6600) = 252 X k, then the value of k
is :
(A) 1650 (B) 1600 (C) 165 (D) 1625
Q17-From the following, the rational number whose decimal expansion is
terminating is
(A)
(B)
(C) (D)
Q18-The decimal expansion of
the rational number will terminate after :
(A) one decimal place (B) two decimal places (C) three decimal places (D) more than 3
decimal places
Q19-The decimal expansion of
the rational number will terminate after :
(A) one decimal place (B) two decimal places (C) three decimal places (D) more than 3
decimal places
Q20-n2 -1 is divisible by 8, if n is
(A) an integer (B) a natural number (C) an odd integer (D) an even integer
Q21-If d=LCM(36, 198), then the value of d is :
(A) 396 (B) 198 (C) 36 (D) 1
Q22-A rational number between
(A) 1.4 (B) 1.75 (C) 1.45 (D) 1.8
and
Q23-If d=HCF(48, 72), the value of d is :
is :
3
(A) 24 (B) 48 (C) 12 (D) 72
Q24-Given that LCM(91, 26)=182, then HCF(91, 26) is :
(A) 13 (B) 26 (C) 7 (D) 9
Q25-H.C.F. of two co–prime numbers is :
(A) 1 (B) 2 (C) 0 (D) 3
Q26-The product of the HCF and LCM of the smallest prime number and the smallest
composite number is :
(A) 2 (B) 4 (C) 6 (D) 8
Q27-The HCF and LCM of two positive integers are ‘h’ and ‘l’ respectively, if one
integer is a, then other will be :
(A)
(B)
(C)
(D)
Q28-1192-1112 is :
(A) Prime number (B) Composite number (C) An odd prime number (D) An odd
composite number
Q29-If HCF (a, 8) = 4, LCM (a, 8) = 24, then a is :
(A) 8 (B) 10 (C) 12 (D) 14
Q30-The decimal expansion of
the rational number will terminate after :
(A) 3 places (B) 4 places (C) 5 places (D) 1 place
Q31-Which of the following is rational ?
(A)
(B)
(C)
(D)
Q32-If m=dn+r, where m, n are positive integers and d and r are integers, then n is
the H.C.F of (m, n) if.
(A) r =1 (B) 0 < r
1 (C) r=0 (D) r is a real number
Q33-A pair of irrational numbers whose product is a rational number is :
(A)
(B)
(C)
(D)
Q34-The decimal expansion of is :
(A) terminating (B) non-terminating and non-recurring (C) non-terminating and
recurring (D) doesn’t exist
4
Q35-If the HCF of 55 and 99 is expressible in the form 55 m - 99, then the value of m
is :
(A) 4 (B) 2 (C) 1 (D) 3
Q36-Given that LCM(91, 26)=182, then HCF(91, 26) is :
(A) 13 (B) 26 (C) 7 (D) 9
Q37-The decimal expansion of
the rational number will terminate after
(A) one decimal place (B) two decimal places (C) three decimal places (D) four
decimal place
Q38-For given positive integers ‘a’ and ‘b’ if a=bq+r ; 0
r < b then :
(A) HCF (a, b)=HCF (b, r) (B) HCF (a, b)=HCF (a, r) (C) HCF (a, q)=HCF (a, r)
(D) HCF (b, q)=HCF (b, r)
Q39-Prove that 7 - 2 2 is irrational.
Let 7 - 2 2 is a rational number
7 -2 2
7=
=
Irrational = rational
Which is not possible
So that 7 - 2 2 is irrational.
Q40-Prove that
is irrational.
Let
is a rational number
Squaring both sides
6 +2 +2
2
=
=
-8
=
Irrational = rational
Which is not possible
So that
is irrational
5
Q41-Prove that
is irrational
Let
is a rational number
Squaring both sides
3 +2 +2
2
=
=
-5
=
Irrational = rational
Which is not possible
So that
is irrational
Q42-Prove that
is irrational.
Let
is a rational number then there exist p and q such that
p and q are co-prime
2
5
P2= 5q2
5|p2
5|p ------------------------------------------------------(1)
P=5c for some integer c
P2|25c2
25c2 =5q2
q2=5c2
5|q2
5|q -------------------------------------------------------(2)
So that 5 is a common factor of p and q , but p and q are co-prime i.e. HCF(p,q)=1
This means that our supposition is wrong
is irrational number
Q43-Prove that
Let
is an irrational.
is a rational number
Irrational = rational
Which is not possible
So that
is irrational
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Q44-Prove that
Let
is an irrational
is a rational number
Irrational = rational
Which is not possible
So that
is irrational
Q45-Prove that
Let
-
-
is irrational.
is a rational number
Squaring both sides
= 9 +20 - 12
12
= 29 =
Irrational = rational
Which is not possible
so that -
is irrational number
Q46-Prove that
is irrational
Let
is a rational number
Squaring both sides
9 +5 - 6
-6
=
=
-(14 -
)
=
Irrational = rational
Which is not possible
Q47-Prove that
is irrational
Let
is a rational number
7
Squaring both sides
25 +3 - 10
-10
=
=
-(28 -
)
=
Irrational = rational
Which is not possible
Q48-Prove that 13 + 25 2 is irrational.
Let 13+25 2 is a rational number
13+25 2
- 13
=
=
irrational = rational
Which is not possible
So that 13 + 25 2 is irrational.
Q49-Prove that 5 + 2 3 is irrational.
Let 5+2 3 is a rational number
5+2 3
-5
=
=
irrational = rational
Which is not possible
So that 5+2 3
is irrational.
Q50-Prove that 15 + 17 3 is irrational
Let 15+17 3 is a rational number
5+17 3
- 15
=
=
8
irrational = rational
Which is not possible
So that 5+17 3 is irrational.
Q51-Prove that (3 + 2 5)2 is irrational
Let (3 + 2 5)2 is rational number
(3 + 2 5)2 =
9 +20 +12
12
=
= - 29
=
=
irrational = rational
Which is not possible
So that (3 + 2 5)2 is irrational.
Q52-Prove that
Let
is irrational
is rational number
=
=
=
( rationalization the Dr)
=
=
rational =ir rational
Which is not possible
So that
is irrational.
Q53-Prove that
is irrational
Let 6+ 2 is a rational number
+ 2
-6
=
irrational = rational
Which is not possible
So that 6+ 2
is irrational.
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Q54-Prove that
Let
is irrational
is rational number
irrational = rational
Which is not possible
So that
is irrational
Q55-Prove that
Let
is irrational.
is rational number
=
22-
=
=
=
rational =irrational
Which is not possible
So that
is irrational
Q56-Prove that 2 - 5 3 is irrational.
Let 2 - 5 3 is a rational number
2-5 3
=
=
irrational = rational
Which is not possible
So that 2 - 5 3
is irrational
Q57-Prove that 2 3 - 7 is irrational
10
Let 2 3 - 7 is a rational number
2 3- 7
+7
=
=
irrational = rational
Which is not possible
So that 2 3 - 7 is irrational
Q58-Prove that
is irrational
Let
is a rational number
Squaring both sides
6 +2 +2
2
=
=
(
- 8)
=
=
Irrational = rational
Which is not possible
So that
is irrational
Q60-Prove that
is irrational
Let 12 3 - 41 is a rational number
12 3- 41
+41
=
=
irrational = rational
Which is not possible
So that 12 3 - 41 is irrational
Q61-Prove that
is irrational
Let
is a rational number
11
Squaring both sides
3 +5 +2
2
=
=
(
- 8)
=
=
Irrational = rational
Which is not possible
So that
is irrational
Q62-Prove that
is irrational
Let 3+ 2 is a rational number
3+ 2
-3
=
irrational = rational
Which is not possible
So that 3+ 2 is irrational.
Q63-Prove that
Let
is irrational
is rational number
X
2+
=
rational = irrational
Which is not possible
So that
is irrational
Q64-If x is rational and
is irrational, then prove that (x+
Let x+ is a rational number
x+
)is irrational.
12
Squaring both sides
x +y +2
2
=
=
(
– x -y)
=
=
Irrational = rational
Which is not possible
So that x+ is irrational
Q65-Using fundamental theorem of arithmetic, find the HCF of 26, 51 and 91.
26= 2 x13
51= 3 x 17
91= 7 x13
HCF = 1
Q66- Find the HCF and LCM of 306 and 54. Verify that HCF x LCM = Product of the
two numbers.
54= 2 x3 x3 x3
306= 2 x 3 x 3 x 17
HCF= 2 x 32= 18
LCM= 33 x 2 x 17= 918
HCF x LCM = 1st no. x 2nd no.
18 x918 = 306 x54
16524 = 16524
Q67-Find the LCM and HCF of 15, 18, 45 by the prime factorization method.
15= 3 x5
18 = 2 x 3 x 3
45 = 3 x 3 x 5
HCF= 3
LCM= 2 x 32 x 5= 90
Q68-If d is the HCF of 45 and 27, find x, y satisfying d = 27x+ 45y
45 = 27 x 1 + 18 -------------------(1)
27 = 18 x 1 + 9 --------------------(2)
18 = 9 x2 + 0 ---------------------(3)
HCF = 9
From (2)
9 = 27 – 18 x1
From (1) 18 = 45 – 27 x1
9 = 27 –( 45- 27 x 1)
9 = 27 x2 – 45
13
= 27 x2 + 45 (-1)
= 27 x + 45y
X = 2, y = -1
Q69- Use Euclid’s division algorithm to find the HCF of 10224 and 9648.
10224 = 9648 x 1 + 576
9648 = 576 x 16 + 432
576 = 432 x1 + 144
432= 144 x 3 + 0
HCF = 144
Q70-Find the LCM of 72, 80 and 120 using the fundamental theorem of arithmetic.
72 = 2 x 2x 2 x 3x3
80= 2x2x2x2x5
120= 2x2x2x3x5
LCM= 24 x 32x 5 = 720
Q71-Find the HCF of 867 and 255, using Euclid’s division algorithm.
867= 255 x 3 + 102
255 = 102 x2 +51
102 = 51 x 2 + 0
HCF = 51
Q72-The HCF and LCM of two numbers are 9 and 90 respectively. If one number is
18, find the other.
Let 2nd number = p
HCF x LCM = 1st no. x 2nd no.
9 x 190 = 18 x p
P = 45
Q73- Find the LCM and HCF of 120 and 144 by using fundamental theorem of
Arithmetic.
120= 2 x 2 x 2 x 3 x 5
144= 2x2x2x2x3x3
HCF= 23 x 3 = 24
LCM = 24 x 32 x 5= 720
Q74-Using fundamental theorem of arithmetic, find the HCF of 26, 51 and 91.
26= 2 x13
51= 3 x 17
91= 7 x13
HCF = 1
14
Q75-Write the denominator of
in the form of 2m 5n, where m, n are non-negative
integers. Also write its decimal expansion without actual division.
Dr = 2 x 54 its decimal expansion up to 4 decimal places because highest power in Dr is
4
Q76-An army contingent of 616 members is to march behind an army band of 32
members in a parade. The two groups are to march in the same number of columns.
What is the maximum number of columns in which they can march ?
616 = 2x2x2x7x11
32= 2x2x2x2x2
HCF= 2x2x2 = 8 No. of columns = 8
Q77-An army contingent of 1000 members is to march behind an army band of 56
members in a parade. The two groups are to march in the same number of columns.
What is the maximum number of columns in which they can march ?
1000= 2x2x2x5x5x5
56=2x2x2x7
HCF= 2x2x2=8
No. of columns = 8
Q78-There are 156, 208 and 260 students in Groups A, B, C respectively. Buses are to
be hired to take them for a field trip. Find the minimum number of buses to be hired
if the same number of students should be accommodated in each bus.
HCF of 156, 208 and 260 is 52
So that 52 students will be accommodate in each bus
Total students = 156 +208 + 260 = 624
No. of buses = 624 ÷ 52 = 12 buses
Q79- Explain why 11x13x15x17+17 is a Composite Number.
Since ( 11 x 13 x 15 + 1) x 17
It has more than two factors so it is composite number
Q80-Explain why 5x4x3x2x1+3 is a composite number
Since ( 5 x 4 x 2 x 1+1) x 3
It has more than two factors so it is composite number
Q81-Is 7 x 6 x 5 x 4x 3 x 2 x 1+5 a composite number ? Justify your answer.
Since ( 7 x6x4x3x2x1+1) x5
It has more than two factors so it is composite number
Q82-Is 7X11x13+11 a composite number ? Justify your answer.
15
Since (7x13+1) x11
It has more than two factors so it is composite number
Q83-Is 75x32x5+3 a composite number ? Justify your answer.
Since (75x 3x5+1) x 3
It has more than two factors so it is composite number
Q84-Show that 4n can never end with the digit zero for any natural number n.
4n =(22)n , only prime factor of 4n is 2
5 does not occur in the prime factorization of 4n
So that 4n does not end is zero
Q85-Show that 8n cannot end with the digit zero for any natural number n.
8n =(23)n , only prime factor of 8n is 2
5 does not occur in the prime factorization of 8n
So that 8n does not end is zero
Q86-Check whether 6n can end with the digit 0 for any natural number n ?
6n =(2 x 3)n , prime factor of 6n is 2 and 3
5 does not occur in the prime factorization of 6n
So that 6n does not end is zero
Q87-Check whether 15n can end with the digit 0 for any natural number n ?
15n =(3 x 5)n , prime factor of 15n is 3 and 5
2 does not occur in the prime factorization of 15n
So that 15n does not end is zero
Q88-Show that square of any positive integer is either of the form 3m or (3m +1) for
some integer m.
Let a be any +ve integer
By Euclid’s division lemma
a=bq + r where b = 3
a= 3q +r
0 r <3
a= 3q when r= 0
a= 3q+1 , if r= 1
a= 3q +2 if r = 2
“ a “ be any +ve integer in the form of 3q, 3q+1 and 3q + 2 for some integer q
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Case I- a = 3q
a2 = ( 3q)2 = 3(3q2)
a2 = 3(3q2)
= 3m where m = 3q2
Case II- a = 3q +1
a2 = ( 3q+1)2 = 9q2+6q + 1
a2 = 3(3q2+ 2q) + 1
= 3m+1 where m = 3q2+ 2q
Case III - a = 3q +2
a2 = ( 3q+2)2 = 9q2+12q + 4 = 9q2+12q +3 + 1
a2 = 3(3q2+ 4q + 1) + 1
= 3m+1 where m = 3q2+ 4q+1
Q89-Use Euclid’s division lemma to show that cube of any positive integer is either
of form 9q, 9q+1, or 9q +8 for some integer q.
Let a be any +ve integer
By Euclid’s division lemma
a=bm + r where b = 3
a= 3m +r
0 r <3
a= 3m when r= 0
a= 3m+1 , if r= 1
a= 3m +2 if r = 2
“ a “ be any +ve integer in the form of 3m, 3m+1 and 3m + 2 for some integer m
Case I- a = 3m
a3 = ( 3m)3 = 9(3m3)
= 9q where q = 3m3
Case II- a = 3m +1
a3 = ( 3m+1)3 = 27m3+27m2 + 9m+ 1
a3 = 9(3m3+3m2+m) + 1
= 9q+1 where q = 3m3+3m2+m
Case III - a = 3m +2
a3 = ( 3m+2)3 = 27m3+ 54m2+ 36m+ 8
a3 = 9(3m3+ 6m2 +4m ) + 8
= 9q+8 where q = 3m3+ 6m2 +4m
So that cube of any positive integer is either of form 9q, 9q+1, or 9q +8 for some integer
q.
Q90-Show that, any positive integer is of the form 3q, 3q+1 or 3q+2, where q is some
integer.
Let a be any +ve integer
By Euclid’s division lemma
17
a=bq + r where b = 3
a= 3q +r
0 r <3
a= 3q when r= 0
a= 3q+1 , if r= 1
a= 3q +2 if r = 2
“ a “ be any +ve integer in the form of 3q, 3q+1 and 3q + 2 for some integer q
Q91-Show that any positive odd integer is of the form 4q+1 or 4q+3 where q is a
positive integer.
Let a be any +ve integer
By Euclid’s division lemma
a=bq + r where b = 4
a= 4q +r
0 r <4
a= 4q when r= 0 even for any +ve integer q = 1,2,3 --- ( a= 4, 8,12-----)
a= 4q+1 , if r= 1 odd for any +ve integer q = 1,2,3------( a= 5,9,13-----)
a= 4q +2 if r = 2 even for any +ve integer q = 1,2,3 --- ( a= 6, 10,14 -----)
a= 4q +3 if r = 3 odd for any +ve integer q = 1,2,3 --- ( a= 7, 11,15-----)
any +ve odd integer in the form of 4q+1 and 4q + 3 for some integer q
Q92-Show that every positive even integer is of the form 2q and that every positive
odd integer is of the form 2q+1, where q is some integer.
Let a be any +ve integer
By Euclid’s division lemma
a=bq + r where b = 2
a= 2q +r
0 r <2
a= 2q when r= 0 even for any +ve integer q = 1,2,3 --- ( a= 2, 4,6-----)
a= 2q+1 , if r= 1 odd for any +ve integer q = 1,2,3------( a= 3,5, 7 -----)
so that every even + ve integer in the form of 2q and odd + ve integer in the form of
2q+1
Q93-Show that the square of any positive integer cannot be of the form 5q+2 or 5q+3
for any integer q.
Let a be any +ve integer
By Euclid’s division lemma
a=bm + r where b = 5
a= 5m +r
0 r <5
a= 5m when r= 0
a= 5m+1 , if r= 1
a= 5m +2 if r = 2
18
a= 5m+3 , if r= 3
a= 5m +4 if r = 4
“ a “ be any +ve integer in the form of 5m, 5m+1 , 5m + 2 , 5m+3 ,5m+ 4 for some
integer m
Case I- a = 5m
a2 = ( 5m)2 = 25m2
a2 = 5(5m2)
= 5q , where q = 5m2
Case II- a = 5m +1
a2 = ( 5m+1)2 = 25m2+10m + 1
a2 = 5(5m2+ 10m) + 1
= 5q+1 , where q = 5m2+ 10m
Case III - a = 5m +2
a2 = ( 5m+2)2 = 25m2+10m + 4
a2 = 5(5m2+ 2m) + 4
= 5q+4 , where q = 5m2+ 2m
Case IV- a = 5m +3
a2 = ( 5m+3)2 = 25m2+30m + 9
a2 = 25m2+30m +5 + 4
a2 = 5(5m2+ 6m +1) + 4
= 5q+4 , where q = 5m2+ 6m+1
Case V- a = 5m +4
a2 = ( 5m+4)2 = 25m2+40m + 16
a2 = 25m2+40m +15 + 1
a2 = 5(5m2+ 8m +3) + 1
= 5q+1 , where q = 5m2+ 8m+3
So that square of any +ve integer cannot be in the form of 5q+2 and 5q+3
Q94-Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5 where q
is some integer.
Let a be any +ve integer
By Euclid’s division lemma
a=bq + r where b = 6
a= 6q +r
0 r <6
a= 6q when r= 0 even for any +ve integer q = 1,2,3 --- ( a= 6, 12,18-----)
a= 6q+1 , if r= 1 odd for any +ve integer q = 1,2,3------( a= 7,13,19-----)
a= 6q +2 if r = 2 even for any +ve integer q = 1,2,3 --- ( a= 8, 14,20 -----)
a= 6q +3 if r = 3 odd for any +ve integer q = 1,2,3 --- ( a= 9, 15,21-----)
a= 6q +4 if r = 4 even for any +ve integer q = 1,2,3 --- ( a= 10, 16,22 -----)
a= 6q +5 if r = 5 odd for any +ve integer q = 1,2,3 --- ( a= 11, 17,23-----)
any +ve odd integer in the form of 6q+1 or 6q + 3 or 6q+5 for some integer q
19
Q95-Show that any positive odd integer is of the form 8q + 1 or 8q + 3 or 8q + 5 or 8q
+ 7,
Let a be any +ve integer
By Euclid’s division lemma
a=bq + r where b = 8
a= 8q +r
0 r <8
a= 8q when r= 0 even for any +ve integer q = 1,2,3 --- ( a= 8,16,24-----)
a= 8q+1 , if r= 1 odd for any +ve integer q = 1,2,3------( a= 9,17,25-----)
a= 8q +2 if r = 2 even for any +ve integer q = 1,2,3 --- ( a= 10,18,26 -----)
a= 8q +3 if r = 3 odd for any +ve integer q = 1,2,3 --- ( a= 11,19,27-----)
a= 8q +4 if r = 4 even for any +ve integer q = 1,2,3 --- ( a= 12,20,28 -----)
a= 8q +5 if r = 5 odd for any +ve integer q = 1,2,3 --- ( a= 13,21,29-----)
a= 8q +6 if r = 6 even for any +ve integer q = 1,2,3 --- ( a= 14,22,30 -----)
a= 8q +7 if r = 7 odd for any +ve integer q = 1,2,3 --- ( a= 15,23,31-----)
any +ve odd integer in the form of 8q+1 or 8q + 3 or 8q+5 or 8q+7 for some integer q
Q96-Use Euclid’s division lemma to show that the square of any positive integer is
either of the form 5m, 5m+1 or 5m+4 for some integer m.
Let a be any +ve integer
By Euclid’s division lemma
a=bq + r where b = 5
a= 5q +r
0 r <5
a= 5q when r= 0
a= 5q+1 , if r= 1
a= 5q +2 if r = 2
a= 5q+3 , if r= 3
a= 5q +4 if r = 4
“ a “ be any +ve integer in the form of 5q, 5q+1 , 5q + 2 , 5q+3 ,5q+ 4 for some integer q
Case I- a = 5q
a2 = ( 5q)2 = 25q2
a2 = 5(5q2)
= 5m , where m = 5q2
Case II- a = 5q +1
a2 = ( 5q+1)2 = 25q2+10q + 1
a2 = 5(5q2+ 10q) + 1
= 5m+1 , where m = 5q2+ 10q
Case III - a = 5q +2
a2 = ( 5q+2)2 = 25q2+10q + 4
a2 = 5(5q2+ 2q) + 4
20
= 5m+4 , where m = 5q2+ 2q
Case IV- a = 5q +3
a2 = ( 5q+3)2 = 25q2+30q + 9
a2 = 25q2+30q +5 + 4
a2 = 5(5q2+ 6q +1) + 4
= 5m+4 , where m = 5q2+ 6q+1
Case V- a = 5q +4
a2 = ( 5q+4)2 = 25q2+40q + 16
a2 = 25q2+40q +15 + 1
a2 = 5(5q2+ 8q +3) + 1
= 5m+1 , where m = 5q2+ 8q+3
So that the square of any positive integer is either of the form 5m, 5m+1 or 5m+4 for
some integer m.
Q97-Show that the square of any positive odd integer is of the form 8 m+1, for some integer
m.
Let a be any +ve integer
By Euclid’s division lemma
a=bq + r where b = 8
a= 8q +r
0 r <8
a= 8q when r= 0 even for any +ve integer q = 1,2,3 --- ( a= 8,16,24-----)
a= 8q+1 , if r= 1 odd for any +ve integer q = 1,2,3------( a= 9,17,25-----)
a= 8q +2 if r = 2 even for any +ve integer q = 1,2,3 --- ( a= 10,18,26 -----)
a= 8q +3 if r = 3 odd for any +ve integer q = 1,2,3 --- ( a= 11,19,27-----)
a= 8q +4 if r = 4 even for any +ve integer q = 1,2,3 --- ( a= 12,20,28 -----)
a= 8q +5 if r = 5 odd for any +ve integer q = 1,2,3 --- ( a= 13,21,29-----)
a= 8q +6 if r = 6 even for any +ve integer q = 1,2,3 --- ( a= 14,22,30 -----)
a= 8q +7 if r = 7 odd for any +ve integer q = 1,2,3 --- ( a= 15,23,31-----)
“ a “ be any +ve odd integer in the form of 8q+1 ,8q+3 and 8q+5 for some integer q
Case I- a = 8q +1
a2 = ( 8q+1)2 = 64q2+16q + 1
a2 = 8(8q2+ 2q) + 1
= 8m+1 , where m = 8q2+ 2q
Case II - a = 8q +3
a2 = ( 8q+3)2 = 64q2+48q + 9
= 64q2+48q + 8 + 1
a2 = 8(8q2+ 6q + 1) + 1
= 8m+1 , where m = 8q2+ 6q + 1
Case III - a = 8q +5
a2 = ( 8q+5)2 = 64q2+80q + 25
= 64q2+80q + 24 + 1
21
a2 = 8(8q2+ 10q + 3) + 1
= 8m+1 , where m = 8q2+ 10q + 3
So that the square of any positive odd integer is of the form 8 m+1, for some integer m.
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