Using an pal x-box 360 on a Japanese ntsc tv.

Hello,
I have a UK PAL x-box 360 and a Japanese Sharp Aquos LC-40AE7 TV.

I can play the x-bo on the tv but I can only get a black and white picture.

I heard that if I connect my x-box to the tv via a HDMI cable, I will get colour as it shold be.
Sadly, I cant test this as my x-box 360 is one of the older models without a hdmi port. However, I will be back in the UK shortly and I will buy a newer x-box 360 with HDMI capability if I can play it on my Japanese tv.
Is using HDMI the solution?
Thaks in advance for any help you can give me.