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Simplified Squaring.

By Sergey M. Nesterishin PE Adjunct Math Instructor ButlerCC

a2=(a-b)(a+b)+b2 Where: a- integer, b -last digit of integer

Examples:

112 =11x11=(11-1)x(11+1)+12=10x12+1=121
122 =12x12=(12-2)x(12+2)+22=10x14+4=144
182 =18x18=(18-8)x(18+8)+82=10x26+64=324
192 =19x19=(19-9)x(19+9)+92=10x28+81=361
212 =21x21=(21-2)x(21+2)+22=20x22+1=441
222 =22x22=(22-2)x(22+2)+22=20x24+4=484
282 =28x28=(28-8)x(28+8)+82=20x36+64=784
292 =29x29=(29-9)x(29+9)+92=20x38+81=841
312 =31x31=(31-2)x(31+2)+22=30x32+1=961
432 =43x43=(43-3)x(43+3)+32=40x46+9=1849


Simplified Multiplication.

a.b=(a-c)(b+c)+c.d Where: a,b- similar integers with different last digits, c,d -last digits of integers

Examples:

24x26 = (24-4)x(26+4)+4x6=20x30+24=624
23x27 = (23-3)x(27+3)+3x7=20x30+21=621
22x28 = (22-2)x(28+2)+2x8=20x30+16=616
21x29 = (21-1)x(29+1)+1x9=20x30+9=609
22x23= (22-2)x(23+2)+2x3=20x25+6=506
22x24 = (22-2)x(24+2)+2x4=20x26+8=528
22x25 = (22-2)x(25+2)+2x5=20x27+10=550
22x26 = (22-2)x(26+2)+2x6=20x28+12=572
23x29 = (23-3)x(29+3)+3x9=20x32+27=667
53x54 = (53-3)x(54+3)+3x4=50x57+12=2862
124x126 = (124-4)x(126+4)+4x6=120x130+24=15624
123x127 = (23-3)x(127+3)+3x7=120x130+21=15621
124x127 = (124-4)x(127+4)+4x7=120x131+28=15748


Divisibility. Two last digits rule.

By Sergey M. Nesterishin PE

Adjunct Math Instructor Butler CC

 

If the sum consisting of two last digits of integer, and number of hundreds in it times reminder from division the one hundred by divisor in question, is divisible by such divisor, then the integer is also divisible by this divisor.

Reminders from division one hundred by numbers 4, 8, and prime numbers from 7 to 47 are: 4~0, 7~2, 8~4, 11~1, 13~9, 17~15, 19~5, 23~8, 29~13, 31~7, 37~26, 41~18, 47~6

Examples:

a=4 b=0 284 84+2x0=84 84:4=21 284:4=71
a=7 b=2 343 43+3x2=49 49:7=7 343:7=49
a=8 b=4 424 24+4x4=44 40:8=5 424:8=53
a=11 b=1 297 97+2x1=99 99:11=9 297:11=27
a=13 b=9 247 47+2x9=65 65:13=5 247:13=19
a=17 b=15 306 06+3x15=51 51:17=3 306:17=18
a=19 b=5 323 23+3x5=38 38:19=2 323:19=17
a=23 b=8 253 53+2x8=69 69:23=3 253:23=11
a=29 b=13 232 32+2x13=58 58:29=2 232:29=8
a=31 b=7 651 51+6x7=93 93:31=3 651:31=21
a=37 b=26 222 22+2x26=74 74:37=2 222:37=6
a=41 b=18 246 46+2x18=82 82:41=2 246:41=6
a=47 b=6 799 99+7x6=141 41+1x6=47 799:47=17
a=7 b=2 2289 89+22x2=133 33+2x1=35 35:7=5
a=13 b=9 4927 27+49x9=468 68+9x4=104 04+9x1=13
a=19 b=5 9139 39+91x5=494 94+4x5=114 14+1x5=19
a=37 b=26 2072 72+20x26=592 92+5x26=222 22+2x26=74
a=47 b=6 1081 81+10x6=141 41+1x6=47 1081:47=23

 

Two last digits divisibility test.
To define whether or not integer is divisible by some particular whole number, perform following operations:

Determine reminder from division one hundred by divisor in question
Multiply reminder by number of hundreds in integer
Add product to two last digits of integer
Proceed with similar operation until integer is reduced to two digits number
Determine by observation is reduced number divisible by divisor in question
If so, non reduced integer is also divisible by such divisor