2.1.5 Special Integers

The following integers have important properties which are exploited regularly on the number sense test. They are:

IntegerProperty
999999=27×37999 = 27 \times 37
7777=10011377 = \frac{1001}{13}
33673367=1010133367 = \frac{10101}{3}
14301430=1001071430 = \frac{10010}{7}
10731073=29×371073 = 29 \times 37
154154=200213154 = \frac{2002}{13}
14431443=1010171443 = \frac{10101}{7}
693693=900913693 = \frac{9009}{13}

The following are some examples showing how to use these special numbers:

999 Trick: 333×127×137=13×999×127×137=13×27⋅3727⋅37=13333 \times \frac{1}{27} \times \frac{1}{37} = \frac{1}{3} \times 999 \times \frac{1}{27} \times \frac{1}{37} = \frac{1}{3} \times \frac{27 \cdot 37}{27 \cdot 37} = \frac{1}{3}

1001 Trick: 385×13=77×5×13=100113×5×13=1001×5=5005385 \times 13 = 77 \times 5 \times 13 = \frac{1001}{13} \times 5 \times 13 = 1001 \times 5 = 5005

10101 Trick: 1443×56=101017×56=10101×567=10101×8=808081443 \times 56 = \frac{10101}{7} \times 56 = 10101 \times \frac{56}{7} = 10101 \times 8 = 80808

Problem Set 2.1.5

572×21572 \times 21
237×999\frac{2}{37} \times 999
33.67×1533.67 \times 15
715×35715 \times 35
3367×213367 \times 21
1073÷291073 \div 29
715×28715 \times 28
429×35429 \times 35
63×42963 \times 429
1073÷371073 \div 37
444×537444 \times \frac{5}{37}
63×57263 \times 572
143×49=1001×143 \times 49 = 1001 \times
29×3729 \times 37
42×71542 \times 715
715×98715 \times 98
27×3727 \times 37
715×77715 \times 77
105×715105 \times 715
429×357429 \times 357
14×71514 \times 715
42×42942 \times 429
21×336.721 \times 336.7
36×3.36736 \times 3.367
715×49715 \times 49
33.67×2733.67 \times 27
707×715707 \times 715
429×21429 \times 21
336.7×3.3336.7 \times 3.3
707×429707 \times 429
385×13385 \times 13
111×727111 \times \frac{7}{27}
539×13539 \times 13
666×237666 \times \frac{2}{37}
537×5548\frac{5}{37} \times 5548 *
333×127×137333 \times \frac{1}{27} \times \frac{1}{37}
462×13462 \times 13
999×727×737999 \times \frac{7}{27} \times \frac{7}{37}
6006÷4626006 \div 462
444×437444 \times \frac{4}{37}
770×13770 \times 13
888×437888 \times \frac{4}{37}
666×1627×2437666 \times \frac{16}{27} \times \frac{24}{37}
143×77143 \times 77
143×63143 \times 63
84×42984 \times 429
143×49143 \times 49
444×537444 \times \frac{5}{37}
222×127222 \times \frac{1}{27}
63×14363 \times 143
555×637555 \times \frac{6}{37}
444×127444 \times \frac{1}{27}
143×77143 \times 77
888×437888 \times \frac{4}{37}
666×127666 \times \frac{1}{27}
777×737777 \times \frac{7}{37}
444×227444 \times \frac{2}{27}
999×337999 \times \frac{3}{37}
666×327666 \times \frac{3}{27}
888×2427888 \times \frac{24}{27}
999×127999 \times \frac{1}{27}
143×13×7143 \times 13 \times 7
666×1837666 \times \frac{18}{37}
999×527999 \times \frac{5}{27}
1001×25=143×1001 \times 25 = 143 \times
3×11×13×213 \times 11 \times 13 \times 21
3×5×7×11×133 \times 5 \times 7 \times 11 \times 13