Evaluate the following:
(i) $102 \times 106$
(ii) $109 \times 107$
(iii) $35 \times 37$
(iv) $53 \times 55$
(v) $103 \times 96$
(vi) $34 \times 36$
(vii) $994 \times 1006$

Given:

(i) $102 \times 106$

(ii) $109 \times 107$

(iii) $35 \times 37$

(iv) $53 \times 55$

(v) $103 \times 96$

(vi) $34 \times 36$

(vii) $994 \times 1006$

To do:

We have to find the given products.

Solution:

Here, to find the given products we can use distributive property twice.

Distributive Property:

The distributive property of multiplication states that when a factor is multiplied by the sum or difference of two terms, it is essential to multiply each of the two numbers by the factor, and finally perform the addition or subtraction operation.

$(a+b)(c+d)=a(c+d)+b(c+d)$..............(I)

(i) The given expression is $102 \times 106$

We can write $102$ as $102=100+2$ and $106$ as $106=100+6$

Therefore,

$102 \times 106=(100+2)\times(100+6)$

$102 \times 106=100(100+6)+2(100+6)$

$102 \times 106=100(100)+100(6)+2(100)+2(6)$

$102 \times 106=10000+600+200+12$

$102 \times 106=10812$

(ii) The given expression is $109 \times 107$

We can write $109$ as $109=100+9$ and $107$ as $107=100+7$

Therefore,

$109 \times 107=(100+9)\times(100+7)$

$109 \times 107=100(100+7)+9(100+7)$

$109 \times 107=100(100)+100(7)+9(100)+9(7)$

$109 \times 107=10000+700+900+63$

$109 \times 107=11663$

(iii) The given expression is $35 \times 37$

We can write $35$ as $35=30+5$ and $37$ as $37=30+7$

Therefore,

$35 \times 37=(30+5)\times(30+7)$

$35 \times 37=30(30+7)+5(30+7)$

$35 \times 37=30(30)+30(7)+5(30)+5(7)$

$35 \times 37=900+210+150+35$

$35 \times 37=1295$

(iv) The given expression is $53 \times 55$

We can write $53$ as $53=50+3$ and $55$ as $55=50+5$

Therefore,

$53 \times 55=(50+3)\times(50+5)$

$53 \times 55=50(50+5)+3(50+5)$

$53 \times 55=50(50)+50(5)+3(50)+3(5)$

$53 \times 55=2500+250+150+15$

$53 \times 55=2915$

(v) The given expression is $103 \times 96$

We can write $103$ as $103=100+3$ and $96$ as $96=100-4$

Therefore,

$103 \times 96=(100+3)\times(100-4)$

$103 \times 96=100(100-4)+3(100-4)$

$103 \times 96=100(100)-100(4)+3(100)-3(4)$

$103 \times 96=10000-400+300-12$

$103 \times 96=9888$

(vi) The given expression is $34 \times 36$

We can write $34$ as $34=30+4$ and $36$ as $36=30+6$

Therefore,

$34 \times 36=(30+4)\times(30+6)$

$34 \times 36=30(30+6)+4(30+6)$

$34 \times 36=30(30)+30(6)+4(30)+4(6)$

$34 \times 36=900+180+120+24$

$34 \times 36=1224$

(vii) The given expression is $994 \times 1006$

We can write $994$ as $994=1000-6$ and $1006$ as $1006=1000+6$

Therefore,

$994 \times 1006=(1000-6)\times(1000+6)$

$994 \times 1006=1000(1000+6)-6(1000+6)$

$994 \times 1006=1000(1000)+1000(6)-6(1000)-6(6)$

$994 \times 1006=1000000+6000-6000-36$

$994 \times 1006=999964$